Q: A car of mass 1200 kg is moving at a constant speed of 20 m/s around a flat circular track of radius 50 m. What is the magnitude of the net force required to keep the car moving in its circular path?

$$F_\text{net} = m \frac{v^2$}{r} = 1200 \times \frac{20^2$}{50} = 9600\ \text{N}$$

Q: A human centrifuge has a radius of 6.1 m and can reach a maximum angular speed of 7.0 rad/s. What is the magnitude of the acceleration experienced by an occupant at the maximum speed?

$$a = r \$$omega^2$$ = 6.1 \times 7.0^2$ = 6.1 \times 49 = 298.9\ \text{m/s}^2$$

Q: If you are rotating your arm so that your hand moves in a vertical circle of radius 0.5 m at a speed of 10 m/s, and your hand has a mass of 0.45 kg, what is the magnitude of the force your wrist must exert on your hand at the lowest point?

$$F = m \frac{v^2$}{r} + mg = 0.45 \times \frac{10^2$}{0.5} + 0.45 \times 9.8 = 0.45 \times 200 + 4.41 = 90 + 4.41 = 94.41\ \text{N}$$

Q: A human centrifuge with a radius of 6.1 m is rotating at a frequency of 1.1 Hz. What is the speed of the capsule at the end of the arm?

$$v = 2 \pi r f = 2 \pi \times 6.1 \times 1.1 \approx 42.2\ \text{m/s}$$

Question 1

A car of mass 1200 kg is moving at a constant speed of 20 m/s around a flat circular track of radius 50 m. What is the magnitude of the net force required to keep the car moving in its circular path?

Accepted Answer

$$F_	ext{net} = m \frac{v^2$}{r} = 1200 	imes \frac{20^2$}{50} = 9600\ 	ext{N}$$

Question 2

A human centrifuge has a radius of 6.1 m and can reach a maximum angular speed of 7.0 rad/s. What is the magnitude of the acceleration experienced by an occupant at the maximum speed?

Accepted Answer

$$a = r \$$omega^2$$ = 6.1 	imes 7.0^2$ = 6.1 	imes 49 = 298.9\ 	ext{m/s}^2$$

Question 3

Which of the following best describes the direction of the acceleration vector for an object in uniform circular motion?

Accepted Answer

It is always directed toward the center of the circle.

Question 4

A bucket of water is swung in a vertical circle. At the top of the circle, what force(s) keep the water in the bucket?

Accepted Answer

The net force (gravity plus the normal force from the bucket) acts downward toward the center of the circle.

Question 5

A ball is shot along a quarter-circle wall and emerges from the opening. Which path does it follow immediately after leaving the wall?

Accepted Answer

It moves in a straight line tangent to the point of exit.

Question 6

A person is riding a bicycle at the top of a loop-the-loop. Which statement is correct about the forces acting on the person at the top?

Accepted Answer

The net force on the person is downward, toward the center of the loop.

Question 7

If you are rotating your arm so that your hand moves in a vertical circle of radius 0.5 m at a speed of 10 m/s, and your hand has a mass of 0.45 kg, what is the magnitude of the force your wrist must exert on your hand at the lowest point?

Accepted Answer

$$F = m \frac{v^2$}{r} + mg = 0.45 	imes \frac{10^2$}{0.5} + 0.45 	imes 9.8 = 0.45 	imes 200 + 4.41 = 90 + 4.41 = 94.41\ 	ext{N}$$

Question 8

Which of the following is the correct relationship between linear speed $$v$$ and angular speed $$\omega$$ for a point at a distance $$r$$ from the axis of rotation?

Accepted Answer

$$v = \omega r$$

Question 9

A rotating rigid body has an angular velocity that increases linearly with time. Which of the following statements is true?

Accepted Answer

The angular acceleration is constant.

Question 10

What is the dimension of the radian, the SI unit for measuring angles?

Accepted Answer

It is dimensionless.

Question 11

A human centrifuge with a radius of 6.1 m is rotating at a frequency of 1.1 Hz. What is the speed of the capsule at the end of the arm?

Accepted Answer

$$v = 2 \pi r f = 2 \pi 	imes 6.1 	imes 1.1 \approx 42.2\ 	ext{m/s}$$

Question 12

If the net force acting on an object in circular motion suddenly vanishes, what path will the object follow?

Accepted Answer

It will move in a straight line tangent to the circle at the point where the force vanished.

Question 13

A rotating arm has a length of 2.0 m and rotates at a constant angular speed of 3.0 rad/s. What is the linear speed of the tip of the arm?

Accepted Answer

$$v = \omega r = 3.0 	imes 2.0 = 6.0\ 	ext{m/s}$$

Question 14

You are swinging your arm around while keeping your arm straight. Which of the following statements is correct?

Accepted Answer

The speed of your hand is greater than the speed of your elbow.

Question 15

A rotating rigid body starts from rest and has a constant angular acceleration $$\alpha. $$What is the angular displacement $$	heta$$ after time $$t$$?

Accepted Answer

$$	heta = \frac{1}{2} \alpha t^2$

Question 16

A pilot in a centrifuge experiences an acceleration of $$16g$$ for one minute. Which of the following is most likely to affect the pilot's tolerance to this acceleration?

Accepted Answer

The pilot's cardiovascular health.

Question 17

Which of the following equations correctly relates the period $$T$$ and frequency $$f of an $$object in uniform circular motion?

Accepted Answer

$$f = \frac{1}{T}$$

Question 18

A rotating object has an angular velocity $$\omega$$ and an angular acceleration $$\alpha. $$Which kinematic equation gives the angular velocity after time $$t if $$the initial angular velocity is $$\$$omega_0$$?

Accepted Answer

$$\omega = \$$omega_0$$ + \alpha t$$

Uniform Circular Motion and Human Tolerance to Acceleration

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