AP Physics Assignment – Circular Motion and Universal Gravitation

Reading            Physics for Scientists and Engineers  – pp. 88 – 86, 137 – 147, 362 – 372

Objectives/HW

 The student will be able to: HW: 1 Solve problems of uniform circular motion involving period, frequency, speed, velocity, acceleration, force. 1 – 10 2 Distinguish, explain, and apply the concepts of centripetal and centrifugal force. 11 – 13 3 Solve problems of uniform circular motion or cycloid motion by use of parametric equations. 14 – 15 4 Solve problems of nonuniform circular motion involving constant rate of change in speed in which there are radial and tangential components of acceleration. 16 – 18 5 State and apply Newton’s Law of Universal Gravitation. 19 – 23 6 Define and apply gravitational field strength. 24 – 28 7 Solve problems involving circular orbits. 29 – 34 8 State, apply, and derive Kepler’s 3rd Law 35 – 36

Homework Problems

1.      (a) Explain why it is impossible to round a curve in your car without accelerating.  (b) There is only one type of curve that involves constant acceleration – what is it and can a car perform such a curve?  (c) Explain why any circular curve will not be constant acceleration.

2.      David puts a 0.85 kg rock in his sling and twirls it at 3.0 Hz.  The rock moves in a circle with radius 35.0 cm.  Ignoring the effect of gravity determine:  (a) the acceleration of the rock, (b) the force that David must exert on the sling, and (c) the speed of the rock when it is released.

3.      A 5.0 g coin falls out of the pocket of a pair of jeans in a washer during the spin cycle.  The cylinder of the washer is rotating at 250 rpm and has diameter 60.0 cm.  The coin “sticks” to the vertical side of the cylinder.  (a) Find the speed of the coin.  (b) Find the acceleration of the coin.  (c) Find the normal force acting on the coin.  (d) What is the minimum value of μs (such that the coin is not sliding down the vertical surface)?

4.      A small block is placed on top of a rotating horizontal platter at a distance r from the center.  The coefficient of static friction is μs.  Derive an expression for the greatest frequency at which the platter can revolve without the block sliding off.

5.      A car is traveling on a highway at 20.0 m/s and encounters a curve in the road during which the direction of the car’s velocity changes by 90.0° in 30.0 s.  (a) Find the car’s centripetal acceleration.  (b) Derive an expression that gives centripetal acceleration in terms of the three variables in this problem:  v, θ, and t.

6.      A mass dangling from the end of a string can be set into motion such that the mass moves in a horizontal circle as shown in the diagram.  The string traces out an imaginary cone; this arrangement is called a conical pendulum.  (a) Use an analysis of the forces acting on the mass in order to show that the acceleration is given by: a = g(r/h), where h is the height of the cone.  (b) Derive and simplify an expression for the period of the motion.

7.      A certain 2130 kg car has a skid-pad rating of 0.85 g.  Analyze this car turning on level pavement:  (a) What is the maximum amount of friction acting in a centripetal direction?  (b) What is the maximum speed this car can go around a circle with radius 50.0 m?  (c) What is the minimum amount of time it can make a U-turn and reverse directions when traveling at a speed of 10.0 m/s?

8.      A car of mass 1750 kg goes around a curve of radius 100.0 m banked at an angle of 10.0° above horizontal.  (a) At what speed could the car complete this curve without the aid of lateral friction?  (c) Assuming μs = 0.90 determine the maximum speed at which the car can go around the same curve.

9.      A race car of mass 1900 kg has a wing (i.e. spoiler) that generates downward force on the car equal to 15 kN when traveling at 75 m/s.  (a) Using μs = 0.90, determine the minimum radius turn around which this car can travel (on level pavement) at this speed.  (b) Repeat but suppose the wing is removed.

10.  A popular amusement park ride is the Gravitron in which riders are enclosed in a cylinder that spins at 24 rpm.  According to Wikipedia the riders experience 4.0 g’s.  (a) Determine the radius at which the riders are positioned.  (b) Find the speed of the riders.

11.  A kid gets on a carousel and sets a 250 g ball on the floor of the carousel.  The ball is 3.0 m from the center and the carousel’s period is 8.0 s.  After the ride begins the kid releases the ball and sees it roll away from the center of the carousel.  (a) Explain why the ball does this.  (b) Ignoring friction, what is the acceleration of the ball relative to the boy.  (c) What centripetal force would the boy have to exert to prevent the ball from rolling away?

12.  The Earth’s surface is technically not an inertial frame of reference – the surface accelerates due to the Earth’s rotation.  (a) Determine the acceleration of the surface of the Earth along the equator.  (b) Because of this noninertial reference frame there appears to be a slight outward or centrifugal force acting on objects along the equator.  Calculate the “centrifugal force” acting on a person of mass 80.0 kg standing on the equator.  (c) If the Earth rotated faster and the day were shorter this effect would be greater – at what length of day would a person at the equator levitate due to centrifugal force?  (d) Physicists sometimes refer to centrifugal force as a fictitious force – explain what is really going on in this problem in terms of an inertial frame of reference and by referring to real forces and the property of inertia.

13.  A popular idea for future space exploration is to rotate a space station or spacecraft in order to create artificial gravity.  Imagine a space station in the form of a giant wheel with diameter d.  (a) Derive an expression that gives the required period of rotation to produce artificial gravity equivalent to earth’s g.  (b) It is thought that revolution rates greater than 2.0 rpm would cause astronauts to become dizzy – what is the minimum diameter for the space station to avoid this?

14.  An object moves in a circle such that its position in meters is given by the following parametric equations , where t is in seconds.  At t = 2.00 s find:  (a) position, (b) velocity, (c) acceleration.  (d) Find the period of the motion.  (e) Sketch the path of the object in the xy plane and show the vectors found in parts a – c.

15.  A bicycle rolls with constant speed 6.00 m/s along a level roadway.  A rock is caught in the tread of one of its tires.  The tire has radius 33.0 cm and rotates in a counterclockwise direction.  Let t = 0, x = 0, y = 0 be the point when the rock touches the pavement.  (a) Determine a set of parametric equations x(t) and y(t) that describe the motion of the rock relative to the earth – hint:  the rock moves uniformly in a circle relative to the bicycle’s frame.  (b) Find the maximum speed of the rock and the point in time at which it occurs.  (c) Sketch the path followed by the rock and show the velocity and acceleration vectors at various points along the path.

16.  A kid plays with a yo-yo of mass 125 g and twirls it in a vertical circle.  The length of the string is 90.0 cm.  The kid twirls it just fast enough to keep it moving in a complete circle – this results in a centripetal acceleration of 5.00 g at the lowest point.  (a) Find the speed of the yo-yo at the highest point.  (b) Find the speed at the lowest point.  (c) Find the tension in the string at the highest and lowest points.  (d) Sketch the path of the yo-yo and show its velocity and acceleration at various points in its motion.

17.  A roadway passes over a hilltop.  The crest of the hill has radius 30.0 m.  (a) Find the normal force acting on the seat of a 75.0 kg driver that goes over the hill at 15.0 m/s?  (b) At what minimum speed would the driver start to come out of the seat when passing over the hill? (c) If the driver comes up and out of the seat there isn’t a net upward force – explain why the driver would come out of the seat if going too fast.

18.  The speed of a car decreases uniformly from 30.0 m/s to 20.0 m/s as it rounds a curve of radius 150.0 m.  The direction of the car’s motion is changed from west to south as it rounds the curve.  (a) Determine the time for the car to round the curve.  (b) Determine the acceleration of the car at a point halfway through the curve.

19.  Two bowling balls – one 5.0 kg and the other 6.0 kg sit on a rack.  The centers of the two balls are 60.0 cm apart.  (a) Find the force that one exerts on the other.  (b) At what separation would this force be quadrupled?

20.  A weightlifter is able to bench press a weight of 900 N on earth.  (a) What mass would have the same weight on Pluto?  (b) Would the weightlifter be able to bench press this mass on Pluto just like he did the object with equal weight on Earth?  Explain.

21.  A spacecraft coasting from Earth to the Moon will lose speed up until a certain point and then gain speed as it nears the Moon.  (a) Find the acceleration at a point halfway between Earth and Moon.  (b) Find the point at which the spacecraft stops losing speed and starts gaining speed.

22.  As the Moon orbits the Earth it reaches a point directly between the Earth and the Sun.  (a) Determine the net force of gravity on the Moon at this point (coming from Earth and Sun).  (b) In light of the result, explain how it is possible for the Moon to continue orbiting the Earth.

23.  In 1995, the Galileo robotic spacecraft released a probe into Jupiter’s atmosphere.  When traveling at 830 m/s the 300 kg-probe’s main chute deployed and slowed it to 40 m/s in 8.0 seconds.  (a) Find the force of Jupiter’s gravity acting on the probe.  (b) Determine the force that the cords of the chute had to withstand.

24.  (a) Calculate g for the surface of Mercury.  (b) Repeat for Mars.  (c) Explain in words how these values can be so close to one another although Mars is much larger than Mercury.

25.  Find the value of the gravitational field at the surface of the following hypothetical worlds as a multiple of Earth’s gravitational field:  (a) Planet Q: twice Earth’s mass, twice Earth’s diameter, (b) Planet X: half Earth’s mass, half Earth’s diameter, (c) Planet S: one tenth Earth’s density, ten times Earth’s diameter,  (d) Planet M: one tenth Earth’s mass, one half Earth’s diameter, (e) Moon L: one hundredth Earth’s mass, one fourth Earth’s diameter.  (Note:  S, M, and L are close approximations of actual solar system bodies – can you figure out which ones?)

26.  (a) At what altitude above Earth is g = 4.90 m/s2?  (b) At what altitude is g = 2.45 m/s2?  (c) At what altitude is g significantly different (say, g = 9.78 m/s2) than at the surface?

27.  It can be shown that the value of g inside an empty spherical shell is zero at all points inside the shell (no matter how massive the shell).  Suppose the Earth had uniform density.  (a) Use these two ideas to solve for g inside the Earth.  (Inside the Earth, at any point a distance r from the center, only the mass contained in a sphere of radius r has a net gravitational effect.  All mass between r and the surface has a net gravitational effect of zero.)  (b) Sketch a graph of g versus r extending from r = 0 to r = 2RE.

28.  Tidal force is related to the difference in g across a given body.  For example, suppose the Moon is located above the Indian Ocean.  The value of Moon’s gravitational field will be stronger in the Indian Ocean than on the opposite side of the Earth in the Pacific Ocean.  (a) Determine the difference in Moon’s g for these two locations using appropriate info about the two bodies.  (b) This difference would be the acceleration of one ocean relative to the other (ignoring other forces).  Estimate how much the surfaces would move apart in one hour’s time assuming constant acceleration from rest.

29.  The space shuttle typically orbits at altitude 300 km.  (a) Find the value of g at this altitude.  (b) Find the speed of the shuttle in this orbit.  (c) Find the period of the orbit.  (d) Find the pull of gravity on a 70.0 kg astronaut aboard the shuttle in this orbit.  (e) Explain why the astronaut floats about inside the shuttle.  (f) In order to leave orbit and return to earth in what direction should rockets fire and why?

30.  In order to place a satellite or spacecraft into orbit about Earth it is not enough to simply lift the object to the correct altitude.  Besides lifting the object into space, what other purpose do the rocket engines serve in order to initiate an orbit?  Explain.

31.  In Feb. 2015, the spacecraft Dawn is scheduled to arrive at the largest asteroid Ceres (m = 9.43 × 1020 kg, r = 470 km) and enter successive orbits at altitudes of 5900 km, 1300 km, and 700 km.  (a) Find the value of g at each altitude and at the surface of Ceres.  (b) Find the speed needed to orbit at each altitude (relative to Ceres).

32.  From 1992 through 2003 astronomers were able to observe a star orbiting compact radio source Sagittarius A at the center of our galaxy.  The star has an orbit with average radius 1.4 × 1014 m and period 15 years.  (a) Use this information to estimate the mass of Sag. A.  (b) How many times more massive is Sag. A than the Sun?  Astronomers infer that Sag. A is a supermassive black hole (it cannot be seen)!

33.  A news report states that a certain satellite is traveling at a speed of 16000 mph in its orbit about Earth.  (a) Determine its altitude.  (b) Find the number of revolutions it makes per day (24 hours).

34.  A geosynchronous satellite revolves in sync with the surface of the Earth and must have a precise period of 23 hours 56 minutes 4.0 seconds.  (a) What is the altitude of the required orbit?  (b) What speed is necessary to “inject” the satellite into this orbit?

35.  Two planets travel in circular orbits around a star.  Planet A has speed v and planet B has speed 3v.  (a) Find the ratio of the two planets’ orbital radii.  (b) Find the ratio of the two planets’ periods.

36.  The following table shows radius (in multiples of Jupiter’s radius) and period (in Earth days) for the orbits of some of Jupiter’s moons.  (a) Use Kepler’s 3rd Law to complete the table.  (b) What graph based on this data would produce a straight line?  Sketch what this graph would look like (don’t actually have to graph the data).

1.      a.
b.

2.      a. 120 m/s2 toward center
b. 110 N toward center
c. 6.6 m/s

3.      a. 7.9 m/s
b. 210 m/s2 toward center
c. 1.0 N toward center
d. 0.048

4.

5.      a. 1.05 m/s2
b.   (θ in degrees)

6.      a.
b.

7.      a. 18 kN
b. 20 m/s
c. 3.8 s

8.      a. 13.1 m/s
b. 35 m/s

9.      a. 350 m
b. 640 m

10.  a. 6.2 m
b. 16 m/s

11.  a.
b. 1.9 m/s2 away from center
c. 0.46 N

12.  a. 0.034 m/s2 toward center or down
b. 2.7 N
c. 1.4 h

13.  a.
b. 450 m

14.  a. 2.0 m, 106° from origin
b. 6.0 m/s, 16°
c. 18 m/s2, 286°
d. 2.1 s

15.  a.
OR

b. 12 m/s (to the left)
at t = 0.173 s, 0.519 s, 0.864 s, etc
c.

16.  a. 2.97 m/s
b. 6.64 m/s
c. 0; 7.35 N
d.

17.  a. 173 N, up
b. 17.1 m/s
c.

18.  a. 9.42 s
b. 4.46 m/s2, 328.8°

19.  a. 5.6 nN
b. 30.0 cm

20.  a. 1370 kg
b. No! – explain

21.  a. 0.0107 m/s2 toward Earth
b. 346 Mm from Earth
38.3 Mm from Moon

22.  a. 2.39 × 1020 N toward Sun
b.

23.  a. 7400 N, down
b. 37 kN tension

24.  a. 3.70 m/s2
b. 3.72 m/s2
c.

25.  a.
b.
c.
d.
e.

26.  a. 2640 km
b. 6380 km
c. 6.5 km

27.  a.   or
b.

28.  a. 2.21 × 10−6 m/s2
b. 14 m

29.  a. 8.94 m/s2
b. 7730 m/s
c. 90.5 minutes
d. 626 N, down
e.
f.

30.

31.  a. 0.0016 m/s2, 0.020 m/s2, 0.046 m/s2,
0.28 m/s2
b. 99 m/s, 190 m/s, 230 m/s

32.  a. 7.24 × 1036 kg
b. 3.6 million suns!

33.  a. 1400 km
b. 13

34.  a. 35787 km
b. 3074.8 m/s

35.  a.
b.

36.  a. 0.675 days, 239 days, 315 Jup. radii
b. r 3 vs. T 2 is linear with slope 65.4