Kinematics: Circular Motion Mechanics: Forces

Transcription

1 Kinematics: Circular Motion Mechanics: Forces Lana heridan De Anza College Oct 11, 2018

2 Last time projectile trajectory equation projectile examples projectile motion and relative motion

3 Overview circular motion force net force

4 Let us now find the magnitude of the acceleration o Circular diagram motion of the position and velocity vectors in Figure 4 Objects the vector that move representing along an arc ofthe a circle change are saidin toposition be D r for The particle follows a circular path of radius r, part of w undergoing circular motion. r O Top view v 162 CHAPTER 6 APPLICATION OF NEWTON LAW r i v i r qu r f v f Accordi constan directio track, th the car v i t a force th First string v f in you feel string, w the circl ward th a The tension b in the string pulls outward It is possible that such an object moves c on the person s with constant hand and pulls speed. inwardbut on the ball. does it move with constant velocity? T T 6 5 FIGURE 6 12 winging a ball in a circle 1 Left Figure: from erway & Jewett, 9th ed. Right Figure: from Walker. To mak that is inc that it m first: Ho

5 Let us now find the magnitude of the acceleration o Circular diagram motion of the position and velocity vectors in Figure 4 Objects the vector that move representing along an arc ofthe a circle change are saidin toposition be D r for The particle follows a circular path of radius r, part of w undergoing circular motion. r O Top view v 162 CHAPTER 6 APPLICATION OF NEWTON LAW r i v i r qu r f v f Accordi constan directio track, th the car v i t a force th First string v f in you feel string, w the circl ward th a The tension b in the string pulls outward It is possible that such an object moves c on the person s with constant hand and pulls speed. inwardbut on the ball. does it move with constant velocity? No! T T 6 5 FIGURE 6 12 winging a ball in a circle 1 Left Figure: from erway & Jewett, 9th ed. Right Figure: from Walker. To mak that is inc that it m first: Ho

6 Circular Motion How large is the acceleration of the object? It should depend on:

7 Circular Motion How large is the acceleration of the object? It should depend on: the speed of the object - in this case, a higher the speed means a larger acceleration the radius of the path - the tighter the turn, the smaller the radius, the larger the acceleration

8 Centripetal Acceleration Centripetal acceleration The acceleration of an object that follows a circular arc of radius, r, at constant speed v. Its magnitude is a = v 2 r (ee page 71 of textbook for the proof.)

9 be modeled as a particle. If it moves Uniform Circular Motion onstant speed v, the magnitude of its The velocity vector points along a tangent to the circle (4.14) otion is given by is (4.15) (4.16) For uniform circular motion: the radius is constant the speed is constant the magnitude of the acceleration is constant ipetal Acceleration of the Earth AM r a c v Examples: of constan fectly circu form magn nucleus in hydrogen

10 Period Period The time for one complete orbit of an object that follows a circular arc of radius, r, at constant speed v. Its magnitude is T = 2πr v

11 Don t fall into the trap of using position of a rigid ob angles measured in degrees in the object, such as a rotational equations. We can also consider the rate at which the angular coordinate angular position is of changing: the object and the fix y uch identification is lational motion as th,t f which is the origin, x motion that the posit r As the particle in,t i tion in a time inte sweeps out an angle u f placement of the rig Uniform Circular Motion O u i The rate at which th Figure 10.2 A particle on a rotating rigid θ object = moves θ f from θ i to spins rapidly, this d slowly, this displacem along the arc of a circle. In the Then we can define the angular speed, ω, as rates can be quantif time interval Dt 5 t f 2 t i, the radial line of length r moves through an omega) as the ratio o ω angular = dθ displacement wheredu θ 5 isumeasured f 2 u i. val in radians Dt during which t dt x

12 Uniform Circular Motion ω gives the amount by which the angle θ advances in radians, per unit time. Therefore, ω = 2π T where T is the period (time for one revolution).

13 Uniform Circular Motion ω gives the amount by which the angle θ advances in radians, per unit time. Therefore, ω = 2π T where T is the period (time for one revolution). Putting in the expression for T (T = 2πr v ): ω = 2π v 2πr ω = v r

14 Uniform Circular Motion ω gives the amount by which the angle θ advances in radians, per unit time. Therefore, ω = 2π T where T is the period (time for one revolution). Putting in the expression for T (T = 2πr v ): ω = 2π v 2πr ω = v r This gives us another expression for the centripetal acceleration: a = ω 2 r

15 Circular Motion Quick Quiz A particle moves in a circular path of radius r with speed v. It then increases its speed to 2v while traveling along the same circular path. (i) The centripetal acceleration of the particle has changed by what factor? Choose one: A 0.25 B 0.5 C 2 D 4 1 Page 93, erway & Jewett

16 Circular Motion Quick Quiz A particle moves in a circular path of radius r with speed v. It then increases its speed to 2v while traveling along the same circular path. (i) The centripetal acceleration of the particle has changed by what factor? Choose one: A 0.25 B 0.5 C 2 D 4 1 Page 93, erway & Jewett

17 Circular Motion Quick Quiz A particle moves in a circular path of radius r with speed v. It then increases its speed to 2v while traveling along the same circular path. (ii) From the same choices, by what factor has the period of the particle changed? A 0.25 B 0.5 C 2 D 4 1 Page 93, erway & Jewett

18 Circular Motion Quick Quiz A particle moves in a circular path of radius r with speed v. It then increases its speed to 2v while traveling along the same circular path. (ii) From the same choices, by what factor has the period of the particle changed? A 0.25 B 0.5 C 2 D 4 1 Page 93, erway & Jewett

19 Non-Uniform Circular Motion a t 5 ` dv dt ` (4 Path of particle a t a r a a a r a r a t a t a a r is the centripetal acceleration. It changes the direction of the particle s velocity. The tangential acceleration a t speeds up or slows down the particle.

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23 Forces Up until now we have predicted the motion of objects from knowledge of their motional quantities, eg. their initial velocities, accelerations, etc. We did not consider what the causes of this motion might be. We now will think about that.

24 Forces Up until now we have predicted the motion of objects from knowledge of their motional quantities, eg. their initial velocities, accelerations, etc. We did not consider what the causes of this motion might be. We now will think about that. We will understand forces as the cause of changes in the motion of objects.

25 Forces Up until now we have predicted the motion of objects from knowledge of their motional quantities, eg. their initial velocities, accelerations, etc. We did not consider what the causes of this motion might be. We now will think about that. We will understand forces as the cause of changes in the motion of objects. Forces are a push or pull that an object experiences because of an interaction. Forces are vectors.

26 Forces Two types of forces contact forces another object came into contact with the object field forces a kind of interaction between objects without them touching each other

27 Forces Force type examples: The Laws of Motion f orce the Contact forces xed t. a b c Field forces m M q Q Iron N d e f orbit around the Earth. This change in velocity is caused by the gravitational force exerted by the Earth on the Moon. When 1 erway a coiled & Jewett, spring Physics is pulled, foras cientists in Figure and 5.1a, Engineers. the spring stretches. When a

28 Forces are Vectors We typically draw them like this 2 : The block is the object that experiences the forces. There are two forces here, N and W, they are drawn as arrows to indicate their direction. 1 Figure from

29 Forces are Vectors A downward force F 1 elongates the spring 1.00 cm. A downward force F 2 elongates the spring 2.00 cm. When F 1 and F 2 are applied together in the same direction, the spring elongates by 3.00 cm. When F 1 is downward and F 2 is horizontal, the combination of the two forces elongates the spring by 2.24 cm F 1 u F 2 F a F 1 b F 2 c F 1 F 2 d Figure of a forc scale. and its direction is u 5 tan 21 (20.500) Because forces have been experimentally verified to behave as vectors, you must use the rules of vector addition to obtain 1 the net force on an object. Figure from erway & Jewett.

30 Net Force 5.2 Newton s First Law and Inertial Frames Net Force F 1 F 2 When and are applied together in the same direction, the spring elongates by 3.00 cm. the vector sum of all forces acting on an object. F 1 F 2 F net = When is downward and F i is horizontal, the combination i of the two forces elongates the spring by 2.24 cm F 1 u F 2 F In the diagram F = F 1 + F 2.

31 elongates by Net 3.00 Force cm. of the two forces elongates the spring by 2.24 cm F 1 u F 2 F c In the diagram F = F 1 + F 2. F 1 The magnitude of F is Figure 5.2 The vector F 2 F = F 2 of a force is tested with a 1 + F 2 2 = = 2.23 N d The direction of F is 0.500) Because θ = forces tan 1 have (F been experiectors, you must use the rules of vector addition 1 /F 2 ) = 26.6 to scale.

32 ummary circular motion forces net force Homework new: Ch 4 Probs: 57, 59, 67 (circular motion) read ahead in Ch 5