Q: A centrifuge spins test tubes at a radius of $$r = 0.1\,\text{m}$$ with a linear speed of $$v = 100\,\text{m/s}. $$What is the centripetal acceleration experienced by the test tubes?

$$a_c = \frac{v^2$}{r} = \frac{100^2$}{0.1} = 100,000\,\text{m/s}^2$$

Q: If the period $$T of an $$object in uniform circular motion is $$2\,\text{s}$$, what is its frequency $$f$$?

$$f = \frac{1}{T} = \frac{1}{2} = 0.5\,\text{Hz}$$

Q: A car of mass $$m$$ travels around a level curve of radius $$r. If $$the maximum static friction force is $$f_{s,\text{max}}$$, what is the maximum speed $$v$$ the car can travel without spinning out?

$$v = \sqrt{\frac{r f_{s,\text{max}}}{m}}$$

Question 1

In uniform circular motion, what is the direction of the acceleration vector relative to the velocity vector at any point on the circle?

Accepted Answer

The acceleration vector is perpendicular to the velocity vector and points toward the center of the circle.

Question 2

A centrifuge spins test tubes at a radius of $$r = 0.1\,	ext{m}$$ with a linear speed of $$v = 100\,	ext{m/s}. $$What is the centripetal acceleration experienced by the test tubes?

Accepted Answer

$$a_c = \frac{v^2$}{r} = \frac{100^2$}{0.1} = 100,000\,	ext{m/s}^2$$

Question 3

If the period $$T of an $$object in uniform circular motion is $$2\,	ext{s}$$, what is its frequency $$f$$?

Accepted Answer

$$f = \frac{1}{T} = \frac{1}{2} = 0.5\,	ext{Hz}$$

Question 4

A puck of mass $$m_p is $$attached to a string and moves in a circle of radius $$r on a $$frictionless table. The string passes through a hole and is attached to a hanging mass $$m_B. $$What speed $$v$$ must the puck have so that the block remains at rest?

Accepted Answer

$$v = \sqrt{\frac{m_B g r}{m_p}}$$

Question 5

In the playground game of tetherball, what provides the centripetal force for the ball as it moves in a horizontal circle?

Accepted Answer

The horizontal component of the rope's tension.

Question 6

A car of mass $$m$$ travels around a level curve of radius $$r. If $$the maximum static friction force is $$f_{s,	ext{max}}$$, what is the maximum speed $$v$$ the car can travel without spinning out?

Accepted Answer

$$v = \sqrt{\frac{r f_{s,	ext{max}}}{m}}$$

Question 7

For an object in uniform circular motion, which equation correctly relates the tangential speed $$v$$, radius $$r$$, and period $$T$$?

Accepted Answer

$$v = \frac{2\pi r}{T}$$

Question 8

The Moon orbits the Earth at an average distance of $$r = 3.8 	imes 10^8$\,	ext{m}$$ with a period $$T = 2.4 	imes 10^6$\,	ext{s}. $$What is the centripetal acceleration $$a_c of $$the Moon?

Accepted Answer

$$a_c = \frac{4\pi^2$ r}{T^2$}$$

Question 9

In non-uniform circular motion, such as a ball on a string spun in a vertical loop, where is the tension in the string largest?

Accepted Answer

At the bottom of the loop, where both tension and weight act toward the center.

Question 10

A car travels around a banked curve. What is the primary reason banking the curve allows for higher speeds without spinning out?

Accepted Answer

Banking increases the normal force, which provides a horizontal component to help with centripetal acceleration.

Question 11

Which equation gives the magnitude of the centripetal acceleration for an object moving at speed $$v in a $$circle of radius $$r$$?

Accepted Answer

$$a_c = \frac{v^2$}{r}$$

Question 12

If the coefficient of static friction between a car's tires and the road is $$\mu_s$$, what is the minimum radius $$r$$ for a curve that a car can take at speed $$v$$ without spinning out?

Accepted Answer

$$r = \frac{v^2$}{\mu_s g}$$

Question 13

In uniform circular motion, why is the motion considered periodic?

Accepted Answer

Because the object repeats its path after a fixed period $$T.$$

Question 14

A ball is swung in a horizontal circle on a string. If the string breaks, what path will the ball follow?

Accepted Answer

It will move in a straight line tangent to the circle at the point of release.

Question 15

Which of the following statements about velocity and acceleration in uniform circular motion is correct?

Accepted Answer

Velocity is always tangent to the circle, and acceleration is always perpendicular to velocity, pointing toward the center.

Question 16

A mass $$m is $$attached to a string and swung in a vertical circle. At the top of the circle, what is the minimum speed required to keep the string taut?

Accepted Answer

$$v_{	ext{min}} = \sqrt{g r}$$

Question 17

What is the relationship between the tension $$T in $$the string and the weight $$mg of a $$ball at the bottom of a vertical loop?

Accepted Answer

$$T = m \frac{v^2$}{r} + mg$$

Question 18

Which of the following best describes the effect of a steering wheel on a car's acceleration?

Accepted Answer

Yes, because it changes the direction of the car, producing acceleration.

Question 19

A centrifuge is used to separate blood components. Why does the centrifuge work for this purpose?

Accepted Answer

The high centripetal acceleration causes denser components to move outward, separating them from less dense components.

Circular Motion and Gravity: Uniform and Non-Uniform Circular Motion

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