# Chapter 9 Circular Motion Dynamics

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Chapter 9 Circular Motion Dynamics Chapter 9 Circular Motion Dynamics Introduction Newton s Second Law and Circular Motion Universal Law of Gravitation and the Circular Orbit of the Moon Universal Law of Gravitation Kepler s Third Law and Circular Motion Worked Examples Circular Motion... 6 Example 9. Geosynchronous Orbit... 6 Example 9. Double Star System... 7 Example 9.3 Rotating Objects... 0 Example 9.4 Tension in a Rope... Example 9.5 Object Sliding in a Circular Orbit on the Inside of a Cone... 3 Example 9.6 Coin on a Rotating Turntable... 5 Appendix 9A The Gravitational Field of a Spherical Shell of Matter

2 Chapter 9 Circular Motion Dynamics I shall now recall to mind that the motion of the heavenly bodies is circular, since the motion appropriate to a sphere is rotation in a circle. Nicholas Copernicus 9. Introduction Newton s Second Law and Circular Motion We have already shown that when an object moves in a circular orbit of radius r with angular velocity, it is most convenient to choose polar coordinates to describe the position, velocity and acceleration vectors. In particular, the acceleration vector is given by # d" & a(t) = r \$ % dt ' ( ˆr(t) + r d " dt ˆ"(t). (9..) Then Newton s Second Law, F = m a, can be decomposed into radial ( ˆr -) and tangential ( ˆ -) components # d" & F r = mr \$ % dt ' ( (circular motion), (9..) F = mr d dt (circular motion). (9..3) For the special case of uniform circular motion, d / dt = 0, and so the sum of the tangential components of the force acting on the object must therefore be zero, F = 0 (uniform circular motion). (9..4) 9. Universal Law of Gravitation and the Circular Orbit of the Moon An important example of (approximate) circular motion is the orbit of the Moon around the Earth. We can approximately calculate the time T the Moon takes to complete one circle around the earth (a calculation of great importance to early lunar calendar systems, which became the basis for our current model.) Denote the distance from the moon to the center of the earth by R e, m. Dedicatory Letter to Pope Paul III. 9-

3 Because the Moon moves nearly in a circular orbit with angular speed = " / T it is accelerating towards the Earth. The radial component of the acceleration (centripetal acceleration) is a r = 4" R e, m T. (9..) According to Newton s Second Law, there must be a centripetal force acting on the Moon directed towards the center of the Earth that accounts for this inward acceleration. 9.. Universal Law of Gravitation Newton s Universal Law of Gravitation describes the gravitational force between two bodies and with masses m and m respectively. This force is a radial force (always pointing along the radial line connecting the masses) and the magnitude is proportional to the inverse square of the distance that separates the bodies. Then the force on object due to the gravitational interaction between the bodies is given by, F, = G m m r, ˆr,, (9..) where r, is the distance between the two bodies and ˆr, is the unit vector located at the position of object and pointing from object towards object. The Universal Gravitation Constant is G = 6.67" 0 N # m # kg. Figure 9. shows the direction of the forces on bodies and along with the unit vector ˆr,. Figure 9. Gravitational force of interaction between two bodies Newton realized that there were still some subtleties involved. First, why should the mass of the Earth act as if it were all placed at the center? Newton showed that for a perfect sphere with uniform mass distribution, all the mass may be assumed to be located at the center. (This calculation is difficult and can be found in Appendix 9A to this 9-3

4 chapter.) We assume for the present calculation that the Earth and the Moon are perfect spheres with uniform mass distribution. Second, does this gravitational force between the Earth and the Moon form an action-reaction Third Law pair? When Newton first explained the Moon s motion in 666, he had still not formulated the Third Law, which accounted for the long delay in the publication of the Principia. The link between the concept of force and the concept of an action-reaction pair of forces was the last piece needed to solve the puzzle of the effect of gravity on planetary orbits. Once Newton realized that the gravitational force between any two bodies forms an action-reaction pair, and satisfies both the Universal Law of Gravitation and his newly formulated Third Law, he was able to solve the oldest and most important physics problem of his time, the motion of the planets. The test for the Universal Law of Gravitation was performed through experimental observation of the motion of planets, which turned out to be resoundingly successful. For almost 00 years, Newton s Universal Law was in excellent agreement with observation. A sign of more complicated physics ahead, the first discrepancy only occurred when a slight deviation of the motion of Mercury was experimentally confirmed in 88. The prediction of this deviation was the first success of Einstein s Theory of General Relativity (formulated in 95). We can apply this Universal Law of Gravitation to calculate the period of the Moon s orbit around the Earth. The mass of the Moon is m = kg and the mass 4 of the Earth is m = kg. The distance from the Earth to the Moon is R e, m = m. We show the force diagram in Figure 9.. Figure 9. Gravitational force of moon Newton s Second Law of motion for the radial direction becomes G m m 4" R e, m = m R. (9..3) e, m T We can solve this equation for the period of the orbit, 9-4

5 T 3 4 Re,m =. (9..4) G m Substitute the given values for the radius of the orbit, the mass of the earth, and the universal gravitational constant. The period of the orbit is T = 4 (3.8 "0 8 m) 3 (6.67 "0 # N \$m \$kg # )(5.98 "0 4 kg) =.35"06 s. (9..5) This period is given in days by " day % T = ( s) # \$ s& ' = 7. days. (9..6) This period is called the sidereal month because it is the time that it takes for the Moon to return to a given position with respect to the stars. The actual time T between full moons, called the synodic month (the average period of the Moon s revolution with respect to the sun and is 9.53 days, it may range between 9.7 days and 9.83 days), is longer than the sidereal month because the Earth is traveling around the Sun. So for the next full moon the Moon must travel a little farther than one full circle around the Earth in order to be on the other side of the Earth from the Sun (Figure 9.3). Figure 9.3: Orbital motion between full moons Therefore the time T between consecutive full moons is approximately T T + T where T T / =.3 days. So T 9.5 days. 9.. Kepler s Third Law and Circular Motion The first thing that we notice from the above solution is that the period does not depend on the mass of the Moon. We also notice that the square of the period is proportional to the cube of the distance between the Earth and the Moon, 9-5

6 T 3 4 Re,m =. (9..7) G m This is an example of Kepler s Third Law, of which Newton was aware. This confirmation was convincing evidence to Newton that his Universal Law of Gravitation was the correct mathematical description of the gravitational force law, even though he still could not explain what caused gravity. 9.3 Worked Examples Circular Motion Example 9. Geosynchronous Orbit A geostationary satellite goes around the earth once every 3 hours 56 minutes and 4 seconds, (a sidereal day, shorter than the noon-to-noon solar day of 4 hours) so that its position appears stationary with respect to a ground station. The mass of the earth is m e = kg. The mean radius of the earth is R e = m. The universal constant of gravitation is G = " N # m # kg ". What is the radius of the orbit of a geostationary satellite? Approximately how many earth radii is this distance? Solution: The satellite s motion can be modeled as uniform circular motion. The gravitational force between the earth and the satellite keeps the satellite moving in a circle (In Figure 9.4, the orbit is close to a scale drawing of the orbit). The acceleration of the satellite is directed towards the center of the circle, that is, along the radially inward direction. Figure 9.4 Geostationary satellite orbit (close to a scale drawing of orbit). Choose the origin at the center of the earth, and the unit vector ˆr along the radial direction. This choice of coordinates makes sense in this problem since the direction of acceleration is along the radial direction. 9-6

7 Let r be the position vector of the satellite. The magnitude of r (we denote it as r s ) is the distance of the satellite from the center of the earth, and hence the radius of its circular orbit. Let be the angular velocity of the satellite, and the period is T = / ". The acceleration is directed inward, with magnitude r ; in vector form, s s a = "r ˆr. (9.3.) Apply Newton s Second Law to the satellite for the radial component. The only force in this direction is the gravitational force due to the Earth, F m r = " rˆ grav s s. (9.3.) The inward radial force on the satellite is the gravitational attraction of the earth, Equating the ˆr components, m m " G rˆ = " m r r ˆ. (9.3.3) s e rs s e rs s s s m m G = m r. (9.3.4) s Solving for the radius of orbit of the satellite r s, " G me # rs = \$ % & ' /3. (9.3.5) The period T of the satellite s orbit in seconds is 8664 s and so the angular speed is " T 8664 s # 5 # = = = 7.9\$ 0 s. (9.3.6) Using the values of, G and me in Equation (9.3.5), we determine r s, r 7 s = 4. 0 m = 6.6 Re. (9.3.7) Example 9. Double Star System Consider a double star system under the influence of gravitational force between the stars. Star has mass m and star has mass m. Assume that each star undergoes uniform circular motion such that the stars are always a fixed distance s apart (rotating counterclockwise in Figure 9.5). What is the period of the orbit? 9-7

8 Figure 9.5 Two stars undergoing circular orbits about each other Solution: Because the distance between the two stars doesn t change as orbit about each other there is a central point where the line connected the two objects intersect as the objects move as can be seen in the figure above. (We will see later on in the course that central point is the center of mass of the system.) Choose radial coordinates for each star with origin at that central point. Let ˆr be a unit vector at Star pointing radially away from the center of mass. The position of object is then r = r ˆr, where r is the distance from the central point. Let ˆr be a unit vector at Star pointing radially away from the center of mass. The position of object is then r = r ˆr, where r is the distance from the central point. Because the distance between the two stars is fixed we have that s = r + r. The coordinate system is shown in Figure 9.6 Figure 9.6 Coordinate system for double star orbits 9-8

9 The gravitational force on object is then The gravitational force on object is then F, = Gm m s ˆr. F, = Gm m s ˆr. The force diagrams on the two stars are shown in Figure 9.7. Figure 9.7 Force diagrams on objects and Let denote the magnitude of the angular velocity of each star about the central point. Then Newton s Second Law, F = m a, for Star in the radial direction ˆr is We can solve this for r, G m m s = m r ". m r = G. s Newton s Second Law, F = m a, for Star in the radial direction ˆr is G m m s = m r ". 9-9

10 We can solve this for r, m r = G. s Because s, the distance between the stars, is constant s = r + r = G m s + G m s = G (m + m ) s. Thus the magnitude of the angular velocity is " = G (m + m ) % # \$ s 3 & ', and the period is then T = " = # 4 s 3 & % \$ G(m + m ) ( '. (9.3.8) Note that both masses appear in the above expression for the period unlike the expression for Kepler s Law for circular orbits. Eq. (9..7). The reason is that in the argument leading up to Eq. (9..7), we assumed that m << m, this was equivalent to assuming that the central point was located at the center of the Earth. If we used Eq. (9.3.8) instead we would find that the orbital period for the circular motion of the Earth and moon about each other is T = 4 (3.8 "0 8 m) 3 (6.67 "0 # N \$m \$kg # )(5.98 "0 4 kg "0 kg) =.33"06 s, which is s = 0.7 d shorter than our previous calculation. Example 9.3 Rotating Objects Two objects and of mass m and m are whirling around a shaft with a constant angular velocity. The first object is a distance d from the central axis, and the second object is a distance d from the axis (Figure 9.8). You may ignore the mass of the strings and neglect the effect of gravity. (a) What is the tension in the string between the inner object and the outer object? (b) What is the tension in the string between the shaft and the inner object? 9-0

11 Figure 9.8 Objects attached to a rotating shaft Solution: We begin by drawing separate force diagrams, Figure 9.9a for object and Figure 9.9b for object. (a) (b) Figure 9.9 (a) and 9.9 (b) Free-body force diagrams for objects and Newton s Second Law, F = m a, for the inner object in the radial direction is ˆr : T T = m d". Newton s Second Law, F = m a, for the outer object in the radial direction is ˆr : T = m d ". The tension in the string between the inner object and the outer object is therefore T = m d. Using this result for T in the force equation for the inner object yields 9-

12 m d " T = "m d, which can be solved for the tension in the string between the shaft and the inner object Example 9.4 Tension in a Rope T = d (m + m ). A uniform rope of mass mand length L is attached to shaft that is rotating at constant angular velocity. Find the tension in the rope as a function of distance from the shaft. You may ignore the effect of gravitation. Solution: Divide the rope into small pieces of length r, each of mass m = (m / L)r. Consider the piece located a distance r from the shaft (Figure 9.0). Figure 9.0 Small slice of rotating rope The radial component of the force on that piece is the difference between the tensions evaluated at the sides of the piece, F r = T (r + r) " T (r), (Figure 9.). Figure 9. Free-body force diagram on small slice of rope The piece is accelerating inward with a radial component a r = r". Thus Newton s Second Law becomes F r = "m# r (9.3.9) T (r + "r) T (r) = (m / L)"r r#. Denote the difference in the tension by T = T (r + r) " T (r). After dividing through by r, Eq. (9.3.9) becomes T r = "(m / L) r#. (9.3.0) 9-

13 In the limit as r " 0, Eq. (9.3.0) becomes a differential equation, dt dr = (m / L)" r. (9.3.) From this, we see immediately that the tension decreases with increasing radius. We shall solve this equation by integration T (r) T (L) = r " =r dt # d r " = (m\$ r / L) r" d r" # d r" L r " = L = (m\$ / L)(r L ) = (m\$ / L)(L r ). (9.3.) We use the fact that the tension, in the ideal case, will vanish at the end of the rope, r = L. Thus, T (r) = (m / L)(L " r ). (9.3.3) This last expression shows the expected functional form, in that the tension is largest closest to the shaft, and vanishes at the end of the rope. Example 9.5 Object Sliding in a Circular Orbit on the Inside of a Cone Consider an object of mass m that slides without friction on the inside of a cone moving in a circular orbit with constant speed v 0. The cone makes an angle with respect to a vertical axis. The axis of the cone is vertical and gravity is directed downwards. The apex half-angle of the cone is as shown in Figure 9.. Find the radius of the circular path and the time it takes to complete one circular orbit in terms of the given quantities and g. Figure 9. Object in a circular orbit on inside of a cone Solution: Choose cylindrical coordinates as shown in the above figure. Choose unit vectors ˆr pointing in the radial outward direction and ˆk pointing upwards. The force diagram on the object is shown in Figure

14 Figure 9.3 Free-body force diagram on object The two forces acting on the object are the normal force of the wall on the object and the gravitational force. Then Newton s Second Law in the ˆr -direction becomes and in the ˆk -direction becomes N cos" = mv r N sin " m g = 0. These equations can be re-expressed as We can divide these two equations, yielding This can be solved for the radius, N cos = m v r N sin = m g. N sin N cos = m g mv / r tan = r g v. r = v g tan. The centripetal force in this problem is the vector component of the contact force that is pointing radially inwards, Fcent = N cos = m g cot, where N sin = m g has been used to eliminate N in terms of m, g and. The radius is independent of the mass because the component of the normal force in the vertical direction must balance the gravitational force, and so the normal force is proportional to the mass. 9-4

15 Example 9.6 Coin on a Rotating Turntable A coin of mass m (which you may treat as a point object) lies on a turntable, exactly at the rim, a distance R from the center. The turntable turns at constant angular speed and the coin rides without slipping. Suppose the coefficient of static friction between the turntable and the coin is given by µ. Let g be the gravitational constant. What is the maximum angular speed max such that the coin does not slip? Figure 9.4 Coin on Rotating Turntable Solution: The coin undergoes circular motion at constant speed so it is accelerating inward. The force inward is static friction and at the just slipping point it has reached its maximum value. We can use Newton s Second Law to find the maximum angular speed max. We choose a polar coordinate system and the free-body force diagram is shown in the figure below. Figure 9.5 Free-body force diagram on coin The contact force is given by C = N + f s = N ˆk f sˆr. (9.3.4) The gravitational force is given by F grav = mg ˆk. (9.3.5) Newton s Second Law in the radial direction is given by f s = m R". (9.3.6) 9-5

16 Newton s Second Law, F z = ma z, in the z-direction, noting that the disc is static hence a z = 0, is given by Thus the normal force is N mg = 0. (9.3.7) N = mg. (9.3.8) As increases, the static friction increases in magnitude until at friction reaches its maximum value (noting Eq. (9.3.8)). s max = max and static ( f ) = µ N = µ mg. (9.3.9) At this value the disc slips. Thus substituting this value for the maximum static friction into Eq. (9.3.6) yields µ mg = mr. (9.3.0) We can now solve Eq. (9.3.0) for maximum angular speed max such that the coin does not slip µ g max =. (9.3.) R max 9-6

17 Appendix 9A The Gravitational Field of a Spherical Shell of Matter When analyzing gravitational interactions between extended bodies we assumed that the when calculating the gravitational force between spherical objects with uniform mass distributions we could treat each sphere as a point-like mass located at the center of the sphere and then use the Universal Law of Gravitation to determine the force between the two point-like objects. We shall now justify that assumption. For simplicity we only need to consider the interaction between a spherical object and a point-like mass. We would like to determine the gravitational force on the point-like object of mass m due to the gravitational interaction with a solid uniform sphere of mass m and radius R. In order to determine the force law we shall first consider the interaction between the point-like object and a uniform spherical shell of mass m s and radius R. We will show that: ) The gravitational force acting on a point-like object of mass m located a distance r > R from the center of a uniform spherical shell of mass m s and radius R is the same force that would arise if all the mass of the shell were placed at the center of the shell. ) The gravitational force on an object of mass m placed inside a spherical shell of matter is zero. The force law summarizes these results: " F s, (r) = G m m s \$ ˆr, r > R # r, \$ % 0, r < R where ˆr is the unit vector located at the position of the object and pointing radially away from the center of the shell. For a uniform spherical distribution of matter, we can divide the sphere into thin shells. Then the force between the point-like object and each shell is the same as if all the mass of the shell were placed at the center of the shell. Then we add up all the contributions of the shells (integration), the spherical distribution can be treated as point-like object located at the center of the sphere justifying our assumption. Thus it suffices to analyze the case of the spherical shell. We shall first divide the shell into small area elements and calculate the gravitational force on the point-like object due to one element of the shell and then add the forces due to all these elements via integration. 9-7

18 We begin by choosing a coordinate system. Choose our z -axis to be directed from the center of the sphere to the position of the object, at position r = z ˆk, so that z 0. (Figure 9A. shows the object lying outside the shell with z > R ). Figure 9A. Object lying outside shell with z > R. Choose spherical coordinates as shown in Figure 9A.. Figure 9A. Spherical coordinates For a point on the surface of a sphere of radius r = R, the Cartesian coordinates are related to the spherical coordinates by x = Rsin cos", y = Rsin sin", z = Rcos, (9.A.) 9-8

19 where 0 # # " and 0 # # ". Note that the angle in Figure 9A. and Equations (9.A.) is not the same as that in plane polar coordinates or cylindrical coordinates. The angle is known as the colatitude, the complement of the latitude. We now choose a small area element shown in Figure 9A.3. Figure 9A.3 Infinitesimal area element The infinitesimal area element on the surface of the shell is given by da = Rsin d d". Then the mass dm contained in that element is dm = da = R sin" d" d#. where is the surface mass density given by = m s / 4" R. The gravitational force F dm, m on the object of mass m that lies outside the shell due to the infinitesimal piece of the shell (with mass dm ) is shown in Figure 9A

20 Figure 9A.4 Force on a point-like object due to piece of shell The contribution from the piece with mass dm to the gravitational force on the object of mass m that lies outside the shell has a component pointing in the negative ˆk -direction and a component pointing radially away from the z -axis. By symmetry there is another mass element with the same differential mass d m = dm on the other side of the shell with same co-latitude but with replaced by ± " ; this replacement changes the sign of x and y in Equations (9.A.) but leaves z unchanged. This other mass element produces a gravitational force that exactly cancels the radial component of the force pointing away from the z -axis. Therefore the sum of the forces of these differential mass elements on the object has only a component in the negative ˆk -direction (Figure 9A.5) Figure 9A.5 Symmetric cancellation of components of force 9-0

21 Therefore we need only the z -component vector of the force due to the piece of the shell on the point-like object. Figure 9A.6 Geometry for calculating the force due to piece of shell. From the geometry of the set-up (Figure 9A.6) we see that Thus (d F s, ) z df z ˆk = "G m dm r s cos# ˆk. df z = G m dm cos" = Gm m s cos" sin\$d\$d%. (9.A.) r s 4# r s The integral of the force over the surface is then F z = Gm & =\$ # =\$ dmcos" % % = Gm m s 4\$ & =0 # =0 The -integral is straightforward yielding r s F z = G m dm cos" = Gm m s r s & =\$ # =\$ cos" sin&d&d# % %. (9.A.3) & =0 # =0 r s cos" sin#d# %. (9.A.4) From Figure 9A.6 we can use the law of cosines in two different ways # =\$ # =0 r s 9-

22 r s = R + z R z cos" R = z + r s r s, z cos#. (9.A.5) Differentiating the first expression in (9.A.5), with R and z constant yields, Hence r s, dr s, = R z sin d. sin d = r s, R z dr s,. (9.A.6) (9.A.7) and from the second expression in (9.A.5) we have that cos = (z " R # ) + r zr s % \$ &. s, (9.A.8) We now have everything we need in terms of r s,. For the case when z > R, r s, varies from z + R to z R. Substituting Equations (9.A.7) and (9.A.8) into Eq. (9.A.3) and using the limits for the definite integral yields F z = Gm s m = Gm s m = Gm s m # =\$ cos" sin# % d# # =0 z % z R z + R r s, r ' z R s, & * z R R z '* & ( ) + r s, ( ) ( ) r s, r s, dr s, R z z R dr s, z R ( % + dr z + R r % z + R s, + s, ) +. (9.A.9) No tables should be needed for these; the result is F z = Gm s m = Gm s m ( ) " z R % \$ + r R z r s, ' # \$ s, & ' " z + R R z # zr z+r ( ) + ( z R) R % & (9.A.0) = Gm s m z. 9-

23 For the case when z < R, r s, varies from R + z to R z. Then the integral is F z = Gm s m = Gm s m = 0. ( ) " z R % \$ + r R z r s, ' # \$ s, & ' " + R + z R z # Rz R+z ( ) + ( z R) z % & (9.A.) So we have demonstrated the proposition that for a point-like object located on the z - axis a distance z from the center of a spherical shell, the gravitational force on the point like object is given by " F s, (r) = G m m s \$ ˆk, z > R # z. \$ % 0, z < R This proves the result that the gravitational force inside the shell is zero and the gravitational force outside the shell is equivalent to putting all the mass at the center of the shell. 9-3

### 9.3 Worked Examples Circular Motion

9.3 Worked Examples Circular Motion Example 9.1 Geosynchronous Orbit A geostationary satellite goes around the earth once every 3 hours 56 minutes and 4 seconds, (a sidereal day, shorter than the noon-to-noon

### = constant of gravitation is G = N m 2 kg 2. Your goal is to find the radius of the orbit of a geostationary satellite.

Problem 1 Earth and a Geostationary Satellite (10 points) The earth is spinning about its axis with a period of 3 hours 56 minutes and 4 seconds. The equatorial radius of the earth is 6.38 10 6 m. The

### Circular Motion Dynamics

Circular Motion Dynamics 8.01 W04D2 Today s Reading Assignment: MIT 8.01 Course Notes Chapter 9 Circular Motion Dynamics Sections 9.1-9.2 Announcements Problem Set 3 due Week 5 Tuesday at 9 pm in box outside

### Proficient. a. The gravitational field caused by a. The student is able to approximate a numerical value of the

Unit 6. Circular Motion and Gravitation Name: I have not failed. I've just found 10,000 ways that won't work.-- Thomas Edison Big Idea 1: Objects and systems have properties such as mass and charge. Systems

### Advanced Higher Physics. Rotational motion

Wallace Hall Academy Physics Department Advanced Higher Physics Rotational motion Problems AH Physics: Rotational Motion 1 2013 Data Common Physical Quantities QUANTITY SYMBOL VALUE Gravitational acceleration

### Physics 1100: Uniform Circular Motion & Gravity

Questions: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Physics 1100: Uniform Circular Motion & Gravity 1. In the diagram below, an object travels over a hill, down a valley, and around a loop the loop at constant

### Problem Solving Circular Motion Kinematics Challenge Problem Solutions

Problem Solving Circular Motion Kinematics Challenge Problem Solutions Problem 1 A bead is given a small push at the top of a hoop (position A) and is constrained to slide around a frictionless circular

### Chapter 9 Lecture. Pearson Physics. Gravity and Circular Motion. Prepared by Chris Chiaverina Pearson Education, Inc.

Chapter 9 Lecture Pearson Physics Gravity and Circular Motion Prepared by Chris Chiaverina Chapter Contents Newton's Law of Universal Gravity Applications of Gravity Circular Motion Planetary Motion and

### Circular Motion and Universal Law of Gravitation. 8.01t Oct 4, 2004

Circular Motion and Universal Law of Gravitation 8.01t Oct 4, 2004 Summary: Circular Motion arc length s= Rθ tangential velocity ds v = = dt dθ R = Rω dt 2 d θ 2 dt tangential acceleration a θ = dv θ =

### Radial Acceleration. recall, the direction of the instantaneous velocity vector is tangential to the trajectory

Radial Acceleration recall, the direction of the instantaneous velocity vector is tangential to the trajectory 1 Radial Acceleration recall, the direction of the instantaneous velocity vector is tangential

### Physics 207 Lecture 10. Lecture 10. Employ Newton s Laws in 2D problems with circular motion

Lecture 10 Goals: Employ Newton s Laws in 2D problems with circular motion Assignment: HW5, (Chapters 8 & 9, due 3/4, Wednesday) For Tuesday: Finish reading Chapter 8, start Chapter 9. Physics 207: Lecture

### Lecture Presentation. Chapter 6 Preview Looking Ahead. Chapter 6 Circular Motion, Orbits, and Gravity

Chapter 6 Preview Looking Ahead Lecture Presentation Chapter 6 Circular Motion, Orbits, and Gravity Text: p. 160 Slide 6-2 Chapter 6 Preview Looking Back: Centripetal Acceleration In Section 3.8, you learned

### Exam 2: Equation Summary

MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics 8.01 Physics Fall Term 2012 Exam 2: Equation Summary Newton s Second Law: Force, Mass, Acceleration: Newton s Third Law: Center of Mass: Velocity

### AP Physics QUIZ Gravitation

AP Physics QUIZ Gravitation Name: 1. If F1 is the magnitude of the force exerted by the Earth on a satellite in orbit about the Earth and F2 is the magnitude of the force exerted by the satellite on the

### 12/1/2014. Chapter 5 Circular Motion; Gravitation. Contents of Chapter 5. Contents of Chapter Kinematics of Uniform Circular Motion

Lecture PowerPoints Chapter 5 Physics: Principles with Applications, 7 th edition Giancoli Chapter 5 Circular Motion; Gravitation This work is protected by United States copyright laws and is provided

### Chapter 5 Part 2. Newton s Law of Universal Gravitation, Satellites, and Weightlessness

Chapter 5 Part 2 Newton s Law of Universal Gravitation, Satellites, and Weightlessness Newton s ideas about gravity Newton knew that a force exerted on an object causes an acceleration. Most forces occurred

### Centripetal force keeps an Rotation and Revolution

Centripetal force keeps an object in circular motion. Which moves faster on a merry-go-round, a horse near the outside rail or one near the inside rail? While a hamster rotates its cage about an axis,

### Lesson 9. Luis Anchordoqui. Physics 168. Tuesday, October 24, 17

Lesson 9 Physics 168 1 Static Equilibrium 2 Conditions for Equilibrium An object with forces acting on it but that is not moving is said to be in equilibrium 3 Conditions for Equilibrium (cont d) First

### Why Doesn t the Moon Hit us? In analysis of this question, we ll look at the following things: i. How do we get the acceleration due to gravity out

Why Doesn t the oon Hit us? In analysis of this question, we ll look at the following things: i. How do we get the acceleration due to gravity out of the equation for the force of gravity? ii. How does

### AP Physics Multiple Choice Practice Gravitation

AP Physics Multiple Choice Practice Gravitation 1. Each of five satellites makes a circular orbit about an object that is much more massive than any of the satellites. The mass and orbital radius of each

### Gravitation. Luis Anchordoqui

Gravitation Kepler's law and Newton's Synthesis The nighttime sky with its myriad stars and shinning planets has always fascinated people on Earth. Towards the end of the XVI century the astronomer Tycho

### 7.4 Universal Gravitation

Circular Motion Velocity is a vector quantity, which means that it involves both speed (magnitude) and direction. Therefore an object traveling at a constant speed can still accelerate if the direction

### Circular Motion and Gravitation Notes 1 Centripetal Acceleration and Force

Circular Motion and Gravitation Notes 1 Centripetal Acceleration and Force This unit we will investigate the special case of kinematics and dynamics of objects in uniform circular motion. First let s consider

### Friction is always opposite to the direction of motion.

6. Forces and Motion-II Friction: The resistance between two surfaces when attempting to slide one object across the other. Friction is due to interactions at molecular level where rough edges bond together:

### Version 001 circular and gravitation holland (2383) 1

Version 00 circular and gravitation holland (383) This print-out should have 9 questions. Multiple-choice questions may continue on the next column or page find all choices before answering. AP B 993 MC

### Chapter 21 Rigid Body Dynamics: Rotation and Translation about a Fixed Axis

Chapter 21 Rigid Body Dynamics: Rotation and Translation about a Fixed Axis Chapter 21 Rigid Body Dynamics: Rotation and Translation about a Fixed Axis... 2 21.1 Introduction... 2 21.2 Translational Equation

### MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Exam Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 1) You are standing in a moving bus, facing forward, and you suddenly fall forward as the

### Planetary Mechanics:

Planetary Mechanics: Satellites A satellite is an object or a body that revolves around another body due to the gravitational attraction to the greater mass. Ex: The planets are natural satellites of the

### What path do the longest sparks take after they leave the wand? Today we ll be doing one more new concept before the test on Wednesday.

What path do the longest sparks take after they leave the wand? Today we ll be doing one more new concept before the test on Wednesday. Centripetal Acceleration and Newtonian Gravitation Reminders: 15

### General Physics I. Lecture 7: The Law of Gravity. Prof. WAN, Xin 万歆.

General Physics I Lecture 7: The Law of Gravity Prof. WAN, Xin 万歆 xinwan@zju.edu.cn http://zimp.zju.edu.cn/~xinwan/ Outline Newton's law of universal gravitation Motion of the planets; Kepler's laws Measuring

### CIRCULAR MOTION AND GRAVITATION

CIRCULAR MOTION AND GRAVITATION An object moves in a straight line if the net force on it acts in the direction of motion, or is zero. If the net force acts at an angle to the direction of motion at any

### F = ma. G mm r 2. S center

In the early 17 th century, Kepler discovered the following three laws of planetary motion: 1. The planets orbit around the sun in an ellipse with the sun at one focus. 2. As the planets orbit around the

### PHYSICS. Chapter 8 Lecture FOR SCIENTISTS AND ENGINEERS A STRATEGIC APPROACH 4/E RANDALL D. KNIGHT Pearson Education, Inc.

PHYSICS FOR SCIENTISTS AND ENGINEERS A STRATEGIC APPROACH 4/E Chapter 8 Lecture RANDALL D. KNIGHT Chapter 8. Dynamics II: Motion in a Plane IN THIS CHAPTER, you will learn to solve problems about motion

### Copyright 2010 Pearson Education, Inc. GRAVITY. Chapter 12

GRAVITY Chapter 12 Units of Chapter 12 Newton s Law of Universal Gravitation Gravitational Attraction of Spherical Bodies Kepler s Laws of Orbital Motion Gravitational Potential Energy Energy Conservation

### Multiple Choice (A) (B) (C) (D)

Multiple Choice 1. A ball is fastened to a string and is swung in a vertical circle. When the ball is at the highest point of the circle its velocity and acceleration directions are: (A) (B) (C) (D) 2.

### Practice Test 3. Name: Date: ID: A. Multiple Choice Identify the choice that best completes the statement or answers the question.

Name: Date: _ Practice Test 3 Multiple Choice Identify the choice that best completes the statement or answers the question. 1. A wheel rotates about a fixed axis with an initial angular velocity of 20

### Episode 403: Orbital motion

Episode 40: Orbital motion In this episode, students will learn how to combine concepts learned in the study of circular motion with Newton s Law of Universal Gravitation to understand the (circular) motion

### 5. Forces and Free-Body Diagrams

5. Forces and Free-Body Diagrams A) Overview We will begin by introducing the bulk of the new forces we will use in this course. We will start with the weight of an object, the gravitational force near

### AP Physics 1 Chapter 7 Circular Motion and Gravitation

AP Physics 1 Chapter 7 Circular Motion and Gravitation Chapter 7: Circular Motion and Angular Measure Gravitation Angular Speed and Velocity Uniform Circular Motion and Centripetal Acceleration Angular

### 10/21/2003 PHY Lecture 14 1

Announcements. Second exam scheduled for Oct. 8 th -- practice exams now available -- http://www.wfu.edu/~natalie/f03phy3/extrapractice/. Thursday review of Chapters 9-4 3. Today s lecture Universal law

### Cutnell/Johnson Physics

Cutnell/Johnson Physics Classroom Response System Questions Chapter 5 Dynamics of Uniform Circular Motion Interactive Lecture Questions 5.1.1. An airplane flying at 115 m/s due east makes a gradual turn

### AP Physics-B Universal Gravitation Introduction: Kepler s Laws of Planetary Motion: Newton s Law of Universal Gravitation: Performance Objectives:

AP Physics-B Universal Gravitation Introduction: Astronomy is the oldest science. Practical needs and imagination acted together to give astronomy an early importance. For thousands of years, the motions

Question 8.1: the following: (a) You can shield a charge from electrical forces by putting it inside a hollow conductor. Can you shield a body from the gravitational influence of nearby matter by putting

### Practice Test 3. Multiple Choice Identify the choice that best completes the statement or answers the question.

Practice Test 3 Multiple Choice Identify the choice that best completes the statement or answers the question. 1. A wheel rotates about a fixed axis with an initial angular velocity of 20 rad/s. During

### Nm kg. The magnitude of a gravitational field is known as the gravitational field strength, g. This is defined as the GM

Copyright FIST EDUCATION 011 0430 860 810 Nick Zhang Lecture 7 Gravity and satellites Newton's Law of Universal Gravitation Gravitation is a force of attraction that acts between any two masses. The gravitation

### Chapter 4 Circular Motion and Gravitation Planetary Data Homework # 26

Planetary Data Homework # 26 PLANETARY DATA Mean Distance Mass from Sun Radius Period Planet (kg) (m) (m) (days) Sun 1.99 x 10 30 6.970 x 10 8 Mercury 3.30 x 10 23 5.791 x 10 10 2.439 x 10 6 87.97 Venus

### Concept Question: Normal Force

Concept Question: Normal Force Consider a person standing in an elevator that is accelerating upward. The upward normal force N exerted by the elevator floor on the person is 1. larger than 2. identical

### Newton s Laws of Motion and Gravity ASTR 2110 Sarazin. Space Shuttle

Newton s Laws of Motion and Gravity ASTR 2110 Sarazin Space Shuttle Discussion Session This Week Friday, September 8, 3-4 pm Shorter Discussion Session (end 3:40), followed by: Intro to Astronomy Department

### Circular Motion & Gravitation FR Practice Problems

1) A mass m is attached to a length L of string and hung straight strainght down from a pivot. Small vibrations at the pivot set the mass into circular motion, with the string making an angle θ with the

### Speed/Velocity in a Circle

Speed/Velocity in a Circle Speed is the MAGNITUDE of the velocity. And while the speed may be constant, the VELOCITY is NOT. Since velocity is a vector with BOTH magnitude AND direction, we see that the

### g = Gm / r 2 The Big Idea

The Big Idea Over 2500 years ago Aristotle proposed two laws of physics governing motion. One for Earthly bodies (objects on Earth) that states objects naturally go in straight lines and one for Heavenly

### Can you shield a body from the gravitational influence of nearby matter by putting it inside a hollow sphere or by some other means?

Question 8.1: the following: You can shield a charge from electrical forces by putting it inside a hollow conductor. Can you shield a body from the gravitational influence of nearby matter by putting it

### Circular Motion and Gravitation

Chapter 6 Circular Motion and Gravitation To understand the dynamics of circular motion. To study the application of circular motion as it applies to Newton's law of gravitation. To examine the idea of

### Chapter 5 Circular Motion; Gravitation

Chapter 5 Circular Motion; Gravitation Units of Chapter 5 Kinematics of Uniform Circular Motion Dynamics of Uniform Circular Motion Highway Curves, Banked and Unbanked Nonuniform Circular Motion Centrifugation

### Gravitation. Objectives. The apple and the Moon. Equations 6/2/14. Describe the historical development of the concepts of gravitational force.

Gravitation Objectives Describe the historical development of the concepts of gravitational force. Describe and calculate how the magnitude of the gravitational force between two objects depends on their

### UNIVERSITY OF SASKATCHEWAN Department of Physics and Engineering Physics

UNIVERSITY OF SASKATCHEWAN Department of Physics and Engineering Physics Physics 115.3 Physics and the Universe FINAL EXAMINATION December 14, 013 NAME: (Last) Please Print (Given) Time: 3 hours STUDENT

Chapter 13. Newton s Theory of Gravity The beautiful rings of Saturn consist of countless centimeter-sized ice crystals, all orbiting the planet under the influence of gravity. Chapter Goal: To use Newton

### Force. Chapter Definition: Force W ~ F ~ N r ~

Chapter 3 Force Before we do any real dynamics, we need to define what a force is. I know that you ve already taken at least one physics course and a statics course. So, I know that you have worked with

### 1 Page. Uniform Circular Motion Introduction. Earlier we defined acceleration as being the change in velocity with time:

Uniform Circular Motion Introduction Earlier we defined acceleration as being the change in velocity with time: a=δv/t Until now we have only talked about changes in the magnitude of the acceleration:

### Laws of Motion. Multiple Choice Questions

The module presented herein is a sequel to MCQs on different units in Physics, a few viz. Rotational motion and Oscillations etc. posted earlier. The contents of the documents are intended to give the

### 1 A car moves around a circular path of a constant radius at a constant speed. Which of the following statements is true?

Slide 1 / 30 1 car moves around a circular path of a constant radius at a constant speed. Which of the following statements is true? The car s velocity is constant The car s acceleration is constant The

### = o + t = ot + ½ t 2 = o + 2

Chapters 8-9 Rotational Kinematics and Dynamics Rotational motion Rotational motion refers to the motion of an object or system that spins about an axis. The axis of rotation is the line about which the

### 2010 Pearson Education, Inc. Chapter 4 Making Sense of the Universe: Understanding Motion, Energy, and Gravity

Chapter 4 Making Sense of the Universe: Understanding Motion, Energy, and Gravity 4.1 Describing Motion: Examples from Daily Life Some of the topics we will explore: How do we describe motion? (Speed,

### CAPA due today. Today will finish up with the hinge problem I started on Wednesday. Will start on Gravity. Universal gravitation

CAPA due today. Today will finish up with the hinge problem I started on Wednesday. Will start on Gravity. Universal gravitation Hinge Problem from Wednesday Hinge Problem cont. F x = 0 = F Nx T cosθ F

### SECOND MIDTERM -- REVIEW PROBLEMS

Physics 10 Spring 009 George A. WIllaims SECOND MIDTERM -- REVIEW PROBLEMS A solution set is available on the course web page in pdf format. A data sheet is provided. No solutions for the following problems:

### Projectile and Circular Motion Review Packet

Conceptual Physics Projectile and Circular Motion Review Packet Mr. Zimmerman Textbook Reference: pp. 33-42, 122-135 Key Terms (fill in definitions) projectile - any object that moves through the air or

### Mechanics II. Which of the following relations among the forces W, k, N, and F must be true?

Mechanics II 1. By applying a force F on a block, a person pulls a block along a rough surface at constant velocity v (see Figure below; directions, but not necessarily magnitudes, are indicated). Which

### Uniform Circular Motion

Uniform Circular Motion Introduction Earlier we defined acceleration as being the change in velocity with time: = Until now we have only talked about changes in the magnitude of the acceleration: the speeding

### dt 2 x = r cos(θ) y = r sin(θ) r = x 2 + y 2 tan(θ) = y x A circle = πr 2

v = v i + at a dv dt = d2 x dt 2 A sphere = 4πr 2 x = x i + v i t + 1 2 at2 x = r cos(θ) V sphere = 4 3 πr3 v 2 = v 2 i + 2a x F = ma R = v2 sin(2θ) g y = r sin(θ) r = x 2 + y 2 tan(θ) = y x a c = v2 r

### EXAM #2. ANSWERS ASTR , Spring 2008

EXAM #2. ANSWERS ASTR 1101-001, Spring 2008 1. In Copernicus s heliocentric model of the universe, which of the following astronomical objects was placed in an orbit around the Earth? The Moon 2. In his

### DEVIL CHAPTER 6 TEST REVIEW

IB PHYSICS Name: Period: Date: # Marks: 51 DEVIL PHYSICS BADDEST CLASS ON CAMPUS CHAPTER 6 TEST REVIEW 1. A cyclist rides around a circular track at a uniform speed. Which of the following correctly gives

### Q1. A) 46 m/s B) 21 m/s C) 17 m/s D) 52 m/s E) 82 m/s. Ans: v = ( ( 9 8) ( 98)

Coordinator: Dr. Kunwar S. Wednesday, May 24, 207 Page: Q. A hot-air balloon is ascending (going up) at the rate of 4 m/s and when the balloon is 98 m above the ground a package is dropped from it, vertically

### Contents. Objectives Circular Motion Velocity and Acceleration Examples Accelerating Frames Polar Coordinates Recap. Contents

Physics 121 for Majors Today s Class You will see how motion in a circle is mathematically similar to motion in a straight line. You will learn that there is a centripetal acceleration (and force) and

### Questions on Gravity and Orbits MS

Questions on Gravity and Orbits MS 1. Using the usual symbols write down an equation for (i) Newton s law of gravitation M1M F G R (ii) Coulomb s law Q1Q F K R State one difference and one similarity between

### Universal Gravitation

Universal Gravitation Johannes Kepler Johannes Kepler was a German mathematician, astronomer and astrologer, and key figure in the 17th century Scientific revolution. He is best known for his laws of planetary

### Chapter 13 TAKE HOME TEST (Gravity&Kepler s Laws) 4/23/13. Make sure your calculator is in degrees, show all work, box answers, UNITS!

AP Physics C! name Chapter 13 TAKE HOME TEST (Gravity&Kepler s Laws) 4/23/13 Make sure your calculator is in degrees, show all work, box answers, UNITS! Chapter Notes: 1.) The free fall acceleration on

### Ph1a: Solution to the Final Exam Alejandro Jenkins, Fall 2004

Ph1a: Solution to the Final Exam Alejandro Jenkins, Fall 2004 Problem 1 (10 points) - The Delivery A crate of mass M, which contains an expensive piece of scientific equipment, is being delivered to Caltech.

### Circular Motion Kinematics

Circular Motion Kinematics 8.01 W04D1 Today s Reading Assignment: MIT 8.01 Course Notes Chapter 6 Circular Motion Sections 6.1-6.2 Announcements Math Review Week 4 Tuesday 9-11 pm in 26-152. Next Reading

### Practice. Newton s 3 Laws of Motion. Recall. Forces a push or pull acting on an object; a vector quantity measured in Newtons (kg m/s²)

Practice A car starts from rest and travels upwards along a straight road inclined at an angle of 5 from the horizontal. The length of the road is 450 m and the mass of the car is 800 kg. The speed of

### Projectile Motion. directions simultaneously. deal with is called projectile motion. ! An object may move in both the x and y

Projectile Motion! An object may move in both the x and y directions simultaneously! The form of two-dimensional motion we will deal with is called projectile motion Assumptions of Projectile Motion! The

### Kepler's Laws and Newton's Laws

Kepler's Laws and Newton's Laws Kepler's Laws Johannes Kepler (1571-1630) developed a quantitative description of the motions of the planets in the solar system. The description that he produced is expressed

### Projectile Motion. Conceptual Physics 11 th Edition. Projectile Motion. Projectile Motion. Projectile Motion. This lecture will help you understand:

Conceptual Physics 11 th Edition Projectile motion is a combination of a horizontal component, and Chapter 10: PROJECTILE AND SATELLITE MOTION a vertical component. This lecture will help you understand:

### B C = B 2 + C 2 2BC cosθ = (5.6)(4.8)cos79 = ) The components of vectors B and C are given as follows: B x. = 6.

1) The components of vectors B and C are given as follows: B x = 6.1 C x = 9.8 B y = 5.8 C y = +4.6 The angle between vectors B and C, in degrees, is closest to: A) 162 B) 111 C) 69 D) 18 E) 80 B C = (

Radians & Radius Circumference = 2πr Part s = θ r r s θ θ in radians! 360 o =2π rad = 6.28 rad θ (rad) = π/180 o θ (deg) θ=s/r 1 Angular speed and acceleration θ f ϖ = t ω = f lim Δt 0 θ t i i Δθ Δt =

### Classical mechanics: conservation laws and gravity

Classical mechanics: conservation laws and gravity The homework that would ordinarily have been due today is now due Thursday at midnight. There will be a normal assignment due next Tuesday You should

### Physics 111. Tuesday, November 9, Universal Law Potential Energy Kepler s Laws. density hydrostatic equilibrium Pascal s Principle

ics Tuesday, ember 9, 2004 Ch 12: Ch 15: Gravity Universal Law Potential Energy Kepler s Laws Fluids density hydrostatic equilibrium Pascal s Principle Announcements Wednesday, 8-9 pm in NSC 118/119 Sunday,

### Lecture 13. Gravity in the Solar System

Lecture 13 Gravity in the Solar System Guiding Questions 1. How was the heliocentric model established? What are monumental steps in the history of the heliocentric model? 2. How do Kepler s three laws

### Conceptual Physics 11 th Edition

Conceptual Physics 11 th Edition Chapter 10: PROJECTILE AND SATELLITE MOTION This lecture will help you understand: Projectile Motion Fast-Moving Projectiles Satellites Circular Satellite Orbits Elliptical

### Chapter 4. Dynamics: Newton s Laws of Motion

Chapter 4 Dynamics: Newton s Laws of Motion Types of Forces: An Overview In nature there are two general types of forces, fundamental and nonfundamental. Fundamental Forces -- three have been identified,

### Chapter 17 Two Dimensional Rotational Dynamics

Chapter 17 Two Dimensional Rotational Dynamics 17.1 Introduction... 1 17.2 Vector Product (Cross Product)... 2 17.2.1 Right-hand Rule for the Direction of Vector Product... 3 17.2.2 Properties of the Vector

### Chapter 8, Rotational Equilibrium and Rotational Dynamics. 3. If a net torque is applied to an object, that object will experience:

CHAPTER 8 3. If a net torque is applied to an object, that object will experience: a. a constant angular speed b. an angular acceleration c. a constant moment of inertia d. an increasing moment of inertia

### Practice Problems from Chapters 11-13, for Midterm 2. Physics 11a Fall 2010

Practice Problems from Chapters 11-13, for Midterm 2. Physics 11a Fall 2010 Chapter 11 1. The Ferris wheel shown below is turning at constant speed. Draw and label free-body diagrams showing the forces

### Chapter 4 Dynamics: Newton s Laws of Motion

Chapter 4 Dynamics: Newton s Laws of Motion Force Newton s First Law of Motion Mass Newton s Second Law of Motion Newton s Third Law of Motion Weight the Force of Gravity; and the Normal Force Applications

### Centripetal acceleration

Book page 250-252 cgrahamphysics.com 2016 Centripetal acceleration Acceleration for circular motion Linear acceleration a = v = v u t t For circular motion: Instantaneous velocity is always tangent to

### 2. Equations of Stellar Structure

2. Equations of Stellar Structure We already discussed that the structure of stars is basically governed by three simple laws, namely hyostatic equilibrium, energy transport and energy generation. In this

### Unit 3 Lesson 2 Gravity and the Solar System. Copyright Houghton Mifflin Harcourt Publishing Company

Florida Benchmarks SC.8.N.1.4 Explain how hypotheses are valuable if they lead to further investigations, even if they turn out not to be supported by the data. SC.8.N.1.5 Analyze the methods used to develop

### Basic Physics. Isaac Newton ( ) Topics. Newton s Laws of Motion (2) Newton s Laws of Motion (1) PHYS 1411 Introduction to Astronomy

PHYS 1411 Introduction to Astronomy Basic Physics Chapter 5 Topics Newton s Laws Mass and Weight Work, Energy and Conservation of Energy Rotation, Angular velocity and acceleration Centripetal Force Angular