**AP Physics – Circular Motion and Gravity Example Problems**

1. A toy mouse of mass 2.50 grams is placed on a turntable that rotates 45.0 rpm. The mouse is 12.0 cm from the center of the turntable. (a) Find the frequency and period of the motion. (b) Determine the speed of the mouse. (c) Find the velocity of the mouse at the northern most point in the circle. (d) Find the mouse’s acceleration at the same point. (e) Find the net force.

2. (a) Suppose the frequency in the previous problem is doubled – what would be the effect on all the values? (b) Suppose the radius is doubled but frequency is still 45 rpm – what is the effect?

3. A hovercraft is set into motion in a circular path. Based on its mass, the radius, and the net force, find its speed and period.

4. A ball is placed inside a hollow plastic hemisphere (like a bowl). The ball is set into motion and revolves in a horizontal circle around the inside of the hemisphere such that the normal is tilted 30.0° from vertical. Given the radius of the inside surface of the hemisphere is 0.10 m, find the period of the motion.

5. A man stands on the equator. Given the radius of the Earth and the radius of its orbit determine: (a) the speed of the man relative to the Earth’s axis, (b) the speed of the Earth’s axis relative to the Sun, (c) the speed of the man relative to the Sun at noon and at midnight on a solstice. (d) Find the acceleration of the man for each of the above.

6. A small wooden block is
placed on a turntable rotating with period *T*. Derive an expression for
the greatest distance from the center of the turntable at which the block can
rest without sliding off.

7. A Mini Cooper of mass
1270 kg (including driver) is tested on a 300.0 ft dia. skid pad. Its best
times are 14.37 s, 15.00 s, and 14.56 s. (a) Find the average best speed on
the skid pad. (b) Find the best acceleration as a number of *g*’s. (c)
Find the amount of lateral friction. (d) Estimate the static coefficient of
friction – what assumptions must be made?

8. Determine the ideal
amount of banking for an interstate cloverleaf in terms of *v* and *r*
assuming no friction “assists” in the turning of the car.

9. A centrifuge rotates at
3200 rpm and the bottom end of each test tube is 9.0 cm away from the axis of
rotation. (a) Determine the centrifugal force in *g*’s. (b) If 1.00 gram
of material is in the bottom of the tube what is the actual force acting upon
it?

10. The position of a particle
is given by **r**(*t*) = *A*(cos (*w t*)* ***i** + sin (*w t*) **j**), where *w
*is constant. Show that the particle moves in a circle of radius *A*. Calculate
*d***r**/*dt* and determine the speed of the particle in terms of *A*
and *w*. Determine *d*^{2}**r**/*dt*^{2}
and determine the acceleration of the particle in terms of *A* and *w*. Show that the acceleration is equal
to *v*^{2}/*r* towards the center

11. A car initially moving west at 5.00 m/s makes a U-turn in precisely 3.00 s at which point the velocity is 3.00 m/s east. Assume the car’s speed decreases uniformly. Find the magnitude of the car’s acceleration 1.50 seconds into the turn.

12. A car moves CCW in a circular path with radius 75 m. At the eastern most point of the circle the car has a speed 10.0 m/s but the driver steps on the accelerator, increasing the speed uniformly at 0.75 m/s per second. Determine the time for the car to reach the northern most point of the circle and determine its velocity and acceleration at that point.

13. (a) Determine the force of gravity that a 5.00 kg bowling ball exerts on a 200.0 g mass that is 30.0 cm from its center. (b) What happens to the magnitude of this force if the distance separating the two objects doubles? triples? halves? (c) At what distance would this force be halved? doubled? tripled?

14. “Weigh” the Earth! Measure the force of gravity on an object with known mass. Use the radius of the Earth to solve for its mass.

15. Proposals have been made to
use gravity in order to tug an asteroid “out of its orbit” so that a collision
with Earth can be avoided. Gravity would act like an invisible rope connecting
a “tugboat” spacecraft with the asteroid. Inspect the figures shown below and confirm
the numbers. (a) Solve for the force of attraction between spacecraft and
asteroid. (b) Solve for the thrust of each rocket engine necessary to prevent
the spacecraft from “falling” into the asteroid. (c) Determine acceleration
rate of the asteroid. (d) Find the time required to achieve the required
delta-*v*.

16. (a) At what altitude above the surface of the Earth would you weigh less by 4.45 N supposing you weigh 800.0 N at zero elevation? (b) At what altitude would you weigh one fourth as much?

17. Find the value of *g*
at the altitudes from the previous problem.

18. Find *g* at the surface
of a neutron star with diameter 20.0 km and mass 1.4 times that of the Sun.
What mass would have the same weight as a typical person – 750 N?

19. The Mars Exploration Rover was
literally “dropped” onto Mars. The lander/rover was surrounded by air bags to
protect it. The 544 kg lander/rover was dropped from a height of 12 m. (a)
Find the impact speed. (b) Find the minimum deflection of the air bags if the deceleration
was not to exceed 20 × earth *g*’s. (c) What force would the air bags
have to withstand in this scenario?

20. Find the strength of the
gravitational field at the surface of the following in terms of Earth’s surface
gravity: (a) Planet *A*: half Earth’s mass, same diameter as Earth, (b)
Planet *B*: half Earth’s mass, two third’s Earth’s diameter. (c) Planet *C*:
same density as Earth, twice Earth’s diameter. (d) Planet *D*: four times
Earth’s mass, three times Earth’s diameter. (e) Planet *E*: one fifth
Earth’s density, ten times Earth’s diameter.

21. Derive an expression for
calculating the *change* in *g* across a certain change in elevation,
*h*, in regions near the surface of Earth. (Hint: use a derivative to
approximate this discrete change.) Use the expression to determine the
variation in *g* from the floor to ceiling – a distance of 3.0 m. How
about the variation across the 400.0 m height of a skyscraper?

22. Use the previous result to
analyze tidal force in the vicinity of a black hole of mass 7.24 × 10 ^{36}
kg. Determine the distance from the black hole at which the difference in *g*
would be 9.8 m/s^{2} across a person of height 1.8 m.

23. The Hubble Space Telescope orbits the Earth at an altitude of 565 km. Determine its speed and period.

24. The International Space Station has a circular orbit about the Earth that has a period of 91.5 minutes. Determine its speed, orbital radius, altitude and acceleration.

25. Derive an expression that gives the speed of a satellite in terms of its orbital radius.

26. Determine the mass of the Sun by analyzing the orbit of one of the planets.

27. There are five positions at
which a small object can orbit the Sun in synch with the Earth. These are
called LaGrange the point. Three of these points lie on a line passing through
Sun and Earth. The spacecraft SOHO is located near *L*_{1}.
Determine the location of *L*_{1}.

28. Consider a moon orbiting a
planet. Not only is the moon attracted to the planet, the planet is attracted
to the moon. Accordingly the planet must accelerate! As a result the planet
moves in a small circle opposite the moon – *i.e.* the planet “wobbles”.
The motions of planet and moon are synchronized such that each has the same
period. (a) Use Newton’s Law of Universal Gravitation to solve for *m*_{1}
in terms of *a*_{2} and *m*_{2} in terms of *a*_{1}.
(b) Use kinematics of circular motion to solve for each acceleration in terms
of radius and period. (c) Now solve for the total mass, *m*_{1} +
*m*_{2}, in terms of the period (and eliminate acceleration). (d)
Rearrange the previous expression and solve for period squared. This result is
called Newton’s modification of Kepler’s 3^{rd} Law. (e) Solve for the
period and each radius for the Earth and Moon based on the mass of each and the
separation: *m*_{1} = 5.974 × 10^{24} kg, *m*_{2}
= 7.35 × 10^{22} kg, *x* = 3.84 × 10^{8} m.