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1 Circular/Gravity ~ Learning Guide Name: Instructions: Using a pencil, answer the following questions. The Pre-Reading is marked, based on effort, completeness, and neatness (not accuracy). The rest of the assignment is marked, based on effort, completeness, neatness, and accuracy. Do your best! Pre-Unit Thoughts: Can an object change velocity without changing speed? How? Remember that velocity is a vector quantity. Circular Motion: 1. A pilot is doing the loop-de-loop manoeuvre as shown. It is said that the pilot is more likely to black out (due to blood leaving the brain) at the bottom of the loop as opposed to the top. With the pilot's head is always towards the centre of the loop, explain (with the help of Newton's Laws) why the bottom location of the loop creates the most danger for black-outs. Research - Describe how flight-suits are designed to help prevent black-outs from occurring. Page 1 of 16

2 2. Newton s First Law states that objects will maintain their original motion unless they are acted on by a Net Force. Newton s Second Law states that objects will accelerate in the direction of the Net Force. Use these two laws to explain the driver s experience (what he/she feels) as well as the direction of the car s acceleration at the five locations shown below. You may assume that the driver keeps his/her speed constant throughout (to avoid tickets). One is done for you Location Describe Motion Direction of Acceleration A B C D E Driver wants to fly off at a tangent and therefore squeezes the seat. He/she will feel heavier at this point (normal force is larger than weight) Acceleration is always towards the centre and is therefore up (centripetal) Page 2 of 16

3 3. For each location above draw a free-body diagram showing the forces acting and the direction of the Net Force. If the Net Force is zero state so. Please construct your vectors so that they are roughly to scale. A B C D E 4. Go to Circular Motion > Lab Resources > Media: Space Station and answer the questions below. a. Using Newton's Laws explain exactly how artificial gravity is created in outer space. b. Describe two ways that engineers could increase the artificial gravity felt in this station. c. Draw a sketch of the space station and label where your feet would be. Show where the best spot to place windows might be and justify your choice. d. Describe what would happen if the space-station suddenly stopped rotating. Page 3 of 16

4 Centripetal Forces: 1. Shown below in a top view are six blocks that are sitting at rest on rotating turntables. All of the turntables have the same rotation rate. The masses of the blocks and how far out from the center they sit varies. Specific values of the variable are given in the figures. Rank these blocks, from greatest to least, on the basis of the magnitude of the horizontal forces holding the blocks on the turntables. That is, put first the block that has the largest force holding it on the turntable and put last the block that has the weakest force holding it on the turntable. Greatest Least Or, all of these blocks are held by equal strength forces. Please carefully explain your reasoning. Page 4 of 16

5 2. A 1.50-kg rock is being twirled in a circle on a frictionless surface using a horizontal rope. The radius of the circle is 2.00 m and the rope make 100 revolutions in 1.00 minutes. a. What is the tension in the rope? Show all work. (ans. 329 N) b. The above will break when the tension exceeds 1000 N. What will be the speed of the rock just as the rope breaks? Show all work. (ans m/s) 3. A popular amusement park ride consists of spinning a drum about a vertical axis as shown below. a. Using Newton's Laws, describe how friction is generated vertically in this ride. b. If the walls of the ride were to suddenly vanish, describe the motion of the passengers inside. c. The work done for one complete revolution is zero. Explain why using both the work-energy theorem and the W = F. d equation (you may have to review these concepts from our Energy Unit) Page 5 of 16

6 4. The moon circles the earth once every 27.3 days. We have already determined that the mass of the earth is kg. What is the distance from the center of the earth to the centre of the moon? Show all work. (ans x 10 8 m) 5. Refer back the first question. What is the apparent weight (F N ) of a 75.0-kg driver travelling at 100 km/h a. over the peak of a hill with radius of curvature equal to 500 m (R 1 ). Show all work. (ans. 619 N) b. at the bottom of the hollow of the same radius (R 2 = R 1 )? Show all work. (ans. 851 N) 6. Bicycle racetracks, or velodromes, are banked at the ends. If we ignore friction, and the banking angle is θ =25 a. What will the maximum speed of a bicycle if it is to move around the end of the track at constant radius r = 25m? Be sure to include a free body diagram. Show all work. (ans m/s) b. If the bicyclist goes faster than this value what must he do to compensate? What will happen? If the bicyclist goes slower? Show any equations to help justify your answer. Page 6 of 16

7 c. How slow could the bicyclist ride and still travel in a circle? The radius at the top of the ramp is 25m and at the bottom it is 15 m. Show all work. (ans m/s) 7. Compare and contrast the dynamics of this question to one of a box (of the same mass as the rider/bike) released from rest down a FRICTIONLESS ramp at an angle of the same value. Construct free-body diagrams of both to compare and comment on the direction of the net force. Which scenario has the larger Normal force? Explain why this is. 8. Engineers design roller coasters for excitement. Feelings of weightlessness and high accelerations make a ride great. You may have noticed that the loops in roller-coasters are rarely circular. They are called clothoid loops Explain how this design changes the experience of the passenger at the top and bottom of the loop. In particular, Use your circular motion equations to assist with your explanation. Page 7 of 16

8 9. A favourite swimming hole includes a rope swing from an elevated landing. Upon approaching the swing someone noticed some calculations in the sand. a. Annotate each step of the calculations explaining the Physics concepts involved in each equation. b. Assuming that the calculations are correct, what was the mass of the person in the calculations? (ans: 60 kg) c. Show how one would solve for tension to get T = 3mg. Page 8 of 16

9 10. Mark and Sue are spinning together on a merry-go-round. Mark yells across to Sue Why are you on the outside far from the centre? According to the equation a c = v 2 /r there is an inverse relation between r and the acceleration. Therefore, if r gets smaller the acceleration will be bigger and the ride will be more exciting. Sue does not seem impressed with Mark s abilities in Physics and yells back According to the equation ac = 4πP2 r/t 2 there is a direct relation between r and a c. If I remain on the outside my acceleration will be bigger and the ride will remain exciting. Both equations are valid for centripetal acceleration. Who is correct? What was the flaw in logic that caused the confusion? 11. What is the minimum speed for a rollercoaster to remain in contact with the tracks if it is doing an upside down loop of radius 350 m. Show your work. (ans m/s) Page 9 of 16

10 Gravity: 1. The gravitational field strength, g, on earth is 9.8 N/kg. This is also known as the acceleration of gravity and could have the units of m/s 2. We often say that F g = mg is simply a specific case of F g = GMm/r 2. Show that the gravitational attraction between any mass, m, and earth, M collapses down to F g = m(9.8 N/kg) provided m is on the surface of the earth. In other words show that F g = GMm/r 2 can be simplified to F g = mg for any location on the surface of the earth. 2. Look up the height of Mount Everest and determine the gravitational field strength, g, at this location. Show all work. (ans m/s 2 ) 3. How far above the Earth s surface must we be before g has decreased significantly by dropping 10%? Show all work. (ans. 345 km) Page 10 of 16

11 4. Ted and Alice are mutually attracted to one another in the gravitational sense. If Ted's mass is 80.0 kg and Alice's is 55.0 kg and they are m apart, what is the magnitude of the attractive force on each? Treat both people as spheres. Show all work. (ans. 1.3 x 10-5 N) 5. If any two masses experience a gravitational pull why is it that Ted and Alice are not pulled into one another to make a people planet? 6. There is a location between the Earth and the Moon where the gravitational pull acting on a mass at this point is zero. It is known as the Langrangian Point. Locate where this point would be and label any forces acting on the mass. You need not calculate the precise location but you must justify your choice of location with the appropriate Physics equation(s). 7. The mass of the planet Mercury is kg and its radius is m. What would a 65.0-kg person weigh on Mercury? What is the acceleration due to gravity on Mercury. Show all work. (ans. 241 N, 3.7 N/kg) Page 11 of 16

12 Orbiting Planets and Circular Motion: 1. Calculate the force of gravity between the earth and the moon. Show all work. (ans. 2.0 x N) 2. Given that the moon orbits the earth in 27.3 days compute the centripetal force acting on the moon. How does this value compare to the question above? Account for any differences. (ans. same as above) 3. For orbiting satellites, the mass of the satellite always cancels in our centripetal force equation. Use the equation to justify why it is the satellite s mass, and not the mass of the central planet, that must cancel. 4. Newton s first law states that objects with more mass have stronger tendencies to maintain their original motion (more inertia). Yet out centripetal force equation suggests that all masses will maintain a steady orbit provided that their tangential speed, v, is the correct match for their orbital radius, r. Provide a physical argument that explains why objects with larger inertias (masses) do not fly off. Is there something that counters this increase in inertia? Page 12 of 16

13 5. Our sun orbits around the center of our galaxy (the Milky Way) once every 2.5 x 10 8 years. The radius of the sun s orbit about the center of the galaxy is approximately 3.15 x m. The sun has made only about 20 revolutions since the Earth was formed over 3 billion years ago! The centripetal force keeping our sun in orbit is supplied by the force of gravity. Calculate the mass of the galaxy. Show all work. (ans x kg) 6. Assuming that the typical star contains ten times the mass of our sun, estimate the number of stars in our galaxy. Show all work. (ans. 1.5 x stars) Page 13 of 16

14 7. The brightest four moons of Jupiter were discovered by Galileo with one of his earliest telescopes. These moons, Io, Europa, Ganymede, and Callisto, are called the Galilean moons in his honour. Some of the available data about these moons are given below. MOON r (km) v T (earthyears) Io Europa Ganymede Callisto The radii are from the centre of Jupiter to the centre of the moon in question. One earth year has 365 days. a. Using the data for above, find the mass of Jupiter. Be specific as to which line of data was used and why. Show all work. (ans x kg) b. From the above data, determine the period of Europa, the distance between Jupiter and Ganymede, and the speed of Callisto. Show all work. (ans x 10 5 s, 1.07 x 10 9 m, 8270 m/s) Page 14 of 16

15 Gravitational Potential Energy Revisited: 1. Objects with Gravitational Potential energy have the ability to do work by falling. Where would a mass need to be located in relation to earth to have absolutely zero potential energy? Use the equation to help with your explanation. 2. Refer to Circular Motion & Universal Gravitation > Gravitational Potential Energy Revisited > Work in Space. Explain why can't we use W = mgh f - mgh i to solve this question more quickly. Similarly, why can we not use W = Fd? 3. How much work would a 60.0 kg person do in climbing from the surface of the Earth to the top of a mountain 2.00 km high? a. Use E p = mgh to find your solution. Show all work. (ans x 10 6 J) b. Use E p = -GMm/r (both answers should be close) c. Why do both methods agree in this case? Page 15 of 16

16 4. Repeat the above question for the same person climbing a ladder to the moon. a. Use E p = mgh to find your solution. Show all work. (ans x J) b. Use E p = -GMm/r. Show all work. (ans x 10 9 J) c. Why do both methods disagree in this case? 5. The escape velocity applies to any object regardless of its mass. Even very small particles, such as atmospheric gases, can reach escape velocities. Gas molecules at 20 C move with a velocity of approximately 5000 m/s, which is well below the escape velocity for Earth - Phew - we breath again! The value of a planet s escape velocity is one of the primary reasons why small planetoids, such as the moon, have no atmosphere. Determine the escape velocity for the moon. Will it be able to trap atmospheric gases at 20 C? Show all work. (ans m/s) Page 16 of 16

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