# Worksheet: Centripetal Force

In this worksheet, we will practice analyzing the magnitudes, directions, and sources of forces that act on objects moving along circular paths.

**Q1: **

What is the magnitude of the centripetal force that must act on an object of mass 1.0 kg to make it move along a circular path of diameter 1.0 m, completing a circle every 1.0 s?

**Q2: **

A car with mass 360 kg travels at constant speed along a circular path around a flat roundabout. The radius of the roundabout is 12 m. The car takes a time of 28 s to completely travel around the roundabout.

What is the friction force between the wheels of the car and the surface of the road?

- A81 N
- B54 N
- C5.5 N
- D220 N
- E350 N

What is the coefficient of static friction for the wheels of the car on the surface of the road?

- A0.60
- B0.023
- C0.10
- D0.062
- E0.015

**Q3: **

A spinning disk has a string that holds a small sphere that has a mass of 16 g. The string is suspended from a point at a negligible distance from its center, as shown in the diagram. The disk rotates uniformly around its center with an angular velocity of 24 rad/s. The string makes an angle of with the axis of rotation of the disk.

What is the magnitude of the vertical force exerted on the sphere by the string?

What is the distance between the disk’s axis of rotation and the circular path followed by the sphere?

What is the magnitude of the horizontal force exerted on the sphere by the string?

What is the magnitude of the net force exerted on the sphere by the string?

**Q4: **

A car with a mass of 660 kg drives at constant speed along a smooth circular track, as shown in the diagram. The car follows a path along the center of the track, maintaining a constant distance to the center of its circular path. The angle of the track above the horizontal .

What is the angle from the vertical at which , the reaction force on the car from the surface of the track, acts?

What is the magnitude of the force that acts on the car along ?

How much time does the car take to return to a point along its path?

**Q5: **

A stone that has a mass of 1.6 kg is swung in a vertical circle at a constant angular velocity of 6.1 rad/s. The stone is attached to a uniform rope of length 0.33 m, as shown in the diagram. The length of the rope is the same as the radius of the circle throughout the motion of the stone.

What is the ratio of the maximum force to the minimum force that the rope can apply to the stone? Give your answer to two significant figures.

What is the force that the rope applies to the stone when the rope makes an angle = above the horizontal?

**Q6: **

A ball at the end of a rope of negligible mass moves uniformly
along a circular path with a radius of
0.48 m.
The centripetal acceleration of
the ball is
63 m/s^{2}.
At a point where the rope makes an angle of
above the horizontal, the rope breaks as it moves downward. At this point, the ball
is 1.5 m
vertically above the ground. Find the horizontal distance between the
ball’s position when the rope breaks and its position when it makes
contact with the ground.

**Q7: **

A stone that has a mass of 2.4 kg is swung in a vertical circle at a constant angular velocity of 7.2 rad/s. The stone is attached to a uniform rope of length 0.25 m, as shown in the diagram. The length of the rope is the same as the radius of the circle throughout the motion of the stone.

What is the difference between the magnitude of the force that the rope applies to the stone at point , where the rope points vertically upward, and the magnitude of the force that the rope applies to the stone at point , where the rope points horizontally?

What is the difference between the magnitude of the force that the rope applies to the stone at point , where the rope points vertically upward, and the magnitude of the force that the rope applies to the stone at point , where the rope points vertically downward?

**Q8: **

Which of the lines on the graph correctly shows how the angular velocity of an object varies with the radius of the circular path followed by the object? Assume that the linear velocity of the object is constant.

- APurple
- BGrey
- COrange
- DBlue

**Q9: **

A ball rolls along a horizontal circular path inside a hollow toroidal pipe, as shown in the diagram. The ball has a mass of 125 g. The ball follows a circular path that has a radius of 17.5 cm. The ball travels all the way through the pipe in a time of 0.642 s.

Which of the following provides the centripetal force on the ball?

- ATension in the pipe
- BFriction of the ball with the surface of the pipe
- CNormal reaction force on the ball
- DGravitational force on the ball

What is the magnitude of the centripetal force on the ball?

**Q10: **

On the graph shown, which of the lines correctly shows how the linear speed of a rotating object varies with the radius of the circular path followed by the object? Assume that the centripetal force acting on the object is the same for any radius of the circular path.

- AOrange
- BBlue
- CRed
- DGrey

**Q11: **

A uniform rope is rotated horizontally around one of its ends, as shown in the diagram. The end of the rope opposite to the fixed end returns to its position every 0.65 s. The free end of the rope moves at constant speed from a point to a point .

What is the ratio of the magnitude of the centripetal acceleration of the point to the magnitude of the centripetal acceleration at point ? Give your answer to two significant figures.

What is the ratio of the magnitude of the centripetal acceleration of the point to the magnitude of the centripetal acceleration at point ? Give your answer to two significant figures.

**Q12: **

Which of the lines on the graph correctly shows how the linear speed of an object varies with the radius of the circular path followed by the object? Assume that the angular velocity of the object is constant.

- AYellow
- BOrange
- CGray
- DBlue

**Q13: **

A star exerts a constant magnitude of gravitational force on an orbiting planet, maintaining the planet in a circular orbit. The planet has an angular velocity of 1.25 μrad/s and the radius of its orbit is m. The mass of the planet is kg.

Calculate the force acting on the planet.

- A N
- B N
- C N
- D N
- E N

Calculate the mass of the star. Use a value of N⋅m^{2}/kg^{2} for the
universal gravitational constant.

- A kg
- B kg
- C kg
- D kg
- E kg

Calculate the orbital period of the planet in Earth days, using exactly 24 hours as the length of one day.

**Q14: **

A roller coaster travels along a part of a track that forms
a vertical loop located between two curved tracks, as shown in the diagram. The
curved tracks are circular arcs with an arc radius of 24 m. The top half of the
loop is a semicircular arc with an arc radius of 14 m. A roller coaster car has
a mass of 3,300 kg. At the base of the curved track, the car has a centripetal
acceleration of 24 m/s^{2}. At the top of the loop, the car has a
centripetal acceleration of 16 m/s^{2}.

What is the ratio of the car’s angular velocity at the top of the loop to the car’s angular velocity at the base of the curved track?

What is the ratio of the normal reaction force on the car at the base of the curved track to the normal reaction force on the car at the top of the loop?