Q: A fighter pilot is flying in a circular turn at a speed of $$300\ \mathrm{m/s}. If $$the maximum centripetal acceleration the pilot can withstand is $$9g ($$where $$g = 9.8\ \mathrm{m/s^2$}$$), what is the minimum radius of the turn the pilot can make without passing out?

$$R = \dfrac{v^2$}{a} = \dfrac{(300)^2$}{9 \times 9.8} \approx 1020\ \mathrm{m}$$

Q: A point at the tip of a propeller blade of length $$0.8\ \mathrm{m}$$ spins at $$3500\ \mathrm{rpm}. $$What is the centripetal acceleration at the tip?

$$a = \$$omega^2$$ R = (367\ \mathrm{s$$^{-1}$$})^2$ \times 0.8\ \mathrm{m} = 1.1 \times 10^5$\ \mathrm{m/s^2$}$$

Q: The Space Shuttle orbits Earth at a radius of $$6.7 \times 10^6$\ \mathrm{m}$$ with a period of $$91\ \mathrm{min}. $$What is the centripetal acceleration of an astronaut aboard the Shuttle?

$$a = \$$omega^2$$ R = (0.00115\ \mathrm{s$$^{-1}$$})^2$ \times 6.7 \times 10^6$\ \mathrm{m} = 8.9\ \mathrm{m/s^2$}$$

Q: If the period of revolution for a satellite in circular orbit is doubled, what happens to its angular velocity?

$$\omega = \dfrac{2\pi}{T}$$,\ so\ if\ T\ \text{doubles},\ \omega\ \text{is halved}$

Q: The Moon orbits the Earth at a radius of $$3.84 \times 10^8$\ \mathrm{m}$$ with a period of $$2.36 \times 10^6$\ \mathrm{s}. $$What is the angular velocity of the Moon?

$$\omega = \dfrac{2\pi}{T} = \dfrac{2\pi}{2.36 \times 10^6$} = 2.66 \times 10$$^{-6}$$\ \mathrm{s$$^{-1}$$}$$

Q: A child is riding a merry-go-round with a radius of $$3\ \mathrm{m} at a $$constant angular velocity of $$0.5\ \mathrm{rad/s}. $$What is the child's linear speed?

$$v = \omega R = 0.5 \times 3 = 1.5\ \mathrm{m/s}$$

Question 1

A fighter pilot is flying in a circular turn at a speed of $$300\ \mathrm{m/s}. If $$the maximum centripetal acceleration the pilot can withstand is $$9g ($$where $$g = 9.8\ \mathrm{m/s^2$}$$), what is the minimum radius of the turn the pilot can make without passing out?

Accepted Answer

$$R = \dfrac{v^2$}{a} = \dfrac{(300)^2$}{9 	imes 9.8} \approx 1020\ \mathrm{m}$$

Question 2

A point at the tip of a propeller blade of length $$0.8\ \mathrm{m}$$ spins at $$3500\ \mathrm{rpm}. $$What is the centripetal acceleration at the tip?

Accepted Answer

$$a = \$$omega^2$$ R = (367\ \mathrm{s$$^{-1}$$})^2$ 	imes 0.8\ \mathrm{m} = 1.1 	imes 10^5$\ \mathrm{m/s^2$}$$

Question 3

The Space Shuttle orbits Earth at a radius of $$6.7 	imes 10^6$\ \mathrm{m}$$ with a period of $$91\ \mathrm{min}. $$What is the centripetal acceleration of an astronaut aboard the Shuttle?

Accepted Answer

$$a = \$$omega^2$$ R = (0.00115\ \mathrm{s$$^{-1}$$})^2$ 	imes 6.7 	imes 10^6$\ \mathrm{m} = 8.9\ \mathrm{m/s^2$}$$

Question 4

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

Accepted Answer

It always points toward the center of the circle (centripetal).

Question 5

A car travels around a circular track of radius $$50\ \mathrm{m} at a $$constant speed of $$20\ \mathrm{m/s}. $$What is the magnitude of its centripetal acceleration?

Accepted Answer

$$a = \dfrac{v^2$}{R} = \dfrac{(20)^2$}{50} = 8\ \mathrm{m/s^2$}$$

Question 6

If the period of revolution for a satellite in circular orbit is doubled, what happens to its angular velocity?

Accepted Answer

$$\omega = \dfrac{2\pi}{T}$$,\ so\ if\ T\ 	ext{doubles},\ \omega\ 	ext{is halved}$

Question 7

A ball is attached to a string and swung in a horizontal circle of radius $$2\ \mathrm{m} at a $$constant speed. If the speed is doubled, by what factor does the centripetal acceleration increase?

Accepted Answer

It increases by a factor of 4.

Question 8

The Moon orbits the Earth at a radius of $$3.84 	imes 10^8$\ \mathrm{m}$$ with a period of $$2.36 	imes 10^6$\ \mathrm{s}. $$What is the angular velocity of the Moon?

Accepted Answer

$$\omega = \dfrac{2\pi}{T} = \dfrac{2\pi}{2.36 	imes 10^6$} = 2.66 	imes 10$$^{-6}$$\ \mathrm{s$$^{-1}$$}$$

Question 9

Which of the following is a correct expression for the position of a particle in uniform circular motion in Cartesian coordinates?

Accepted Answer

$$x = R \cos(\omega t),\ y = R \sin(\omega t)$$

Question 10

A satellite moves in a circular orbit with constant speed. Which of the following statements is true?

Accepted Answer

The velocity vector changes direction but not magnitude.

Question 11

If the frequency of revolution of a wheel is $$f$$, what is the angular velocity $$\omega$$?

Accepted Answer

$$\omega = 2\pi f$$

Question 12

A runner completes one lap around a $$400\ \mathrm{m}$$ track in $$50\ \mathrm{s}. $$What is the magnitude of the runner's average velocity?

Accepted Answer

0 $$\mathrm{m/s}$$

Question 13

Which of the following is the correct relationship between period $$T$$ and frequency $$f$$ for uniform circular motion?

Accepted Answer

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

Question 14

A child is riding a merry-go-round with a radius of $$3\ \mathrm{m} at a $$constant angular velocity of $$0.5\ \mathrm{rad/s}. $$What is the child's linear speed?

Accepted Answer

$$v = \omega R = 0.5 	imes 3 = 1.5\ \mathrm{m/s}$$

Question 15

A stone is tied to a string and whirled in a horizontal circle of radius $$1.2\ \mathrm{m} at a $$speed of $$6\ \mathrm{m/s}. $$What is the tension in the string if the stone has a mass of $$0.5\ \mathrm{kg}$$?

Accepted Answer

$$F = m \dfrac{v^2$}{R} = 0.5 	imes \dfrac{36}{1.2} = 15\ \mathrm{N}$$

Uniform Circular Motion: Physics with Calculus Study Guide

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