Exercises: Uniform circular motion

The horses on a merry-go-round always rotate at the same speed, describing circles.

1. a) What type of motion do they describe?
2. b) Explain in your own words what the period and frequency of this motion are.

A wheel completes $50$ turns in $10\ \text{s}$.

1. a) Calculate its period, in seconds.
2. b) Calculate its frequency, in hertz ($\text{Hz}$).

A blender motor rotates at a frequency of $2\ \text{Hz}$.

1. a) What is its period?
2. b) How many turns does it complete in one minute?

Observe the second hand of an analogue clock.

1. a) What is its period, that is, how long does it take to make one complete turn?
2. b) How many turns does it make in one hour?

A Ferris wheel takes $40\ \text{s}$ to complete one full turn.

1. a) How many turns does it complete in $10\ \text{min}$?
2. b) How long does it take to make $6$ turns? Express the result in minutes.

A horse on a merry-go-round rotates at $4\ \text{m}$ from the center, and the merry-go-round takes $10\ \text{s}$ to make one turn, as shown in the figure.

Calculate its linear speed using $v = \frac{2 \pi r}{T}$. Take $\pi = 3.14$ and round the result to one decimal place.

Two children get on the same merry-go-round: one on an inner horse, $2\ \text{m}$ from the center, and another on an outer horse, $5\ \text{m}$ from the center. The diagram is shown in the figure.

1. a) Do they take the same time to complete one turn? What physical quantity do they share?
2. b) Which of the two moves faster? Explain using the distance covered by each in the same amount of time.

A turntable rotates at $45$ revolutions per minute ($\text{rpm}$).

1. a) Express its frequency in hertz (revolutions per second).
2. b) Calculate its period, rounded to one decimal place.

A desk fan makes one turn every $0.1\ \text{s}$ and its blades have a radius of $0.2\ \text{m}$.

1. a) What is its frequency?
2. b) Calculate the linear speed of the tip of a blade using $v = \frac{2 \pi r}{T}$. Take $\pi = 3.14$.

A bicycle travels at $6\ \text{m/s}$ and its wheels have a radius of $0.3\ \text{m}$.

1. a) Calculate the rotational period of the wheels, that is, the time they take to complete one turn. Hint: in one turn, a point on the wheel travels $2 \pi r$.
2. b) Calculate the rotational frequency, in $\text{Hz}$.

Take $\pi = 3.14$ and round the results to one decimal place.

The drum of a washing machine spins at $1200\ \text{rpm}$ and has a radius of $0.25\ \text{m}$.

1. a) Express its frequency in hertz.
2. b) Calculate its period.
3. c) Calculate the linear speed of a point on the edge of the drum. Take $\pi = 3.14$.

A child on a merry-go-round is $3\ \text{m}$ from the center and moves at $1.5\ \text{m/s}$.

1. a) Calculate the period of the merry-go-round, rounded to one decimal place. Take $\pi = 3.14$.
2. b) Calculate its angular velocity using $\omega = \frac{v}{r}$ and state its unit ($\text{rad/s}$).

The Earth rotates on its axis with a period of $24\ \text{h}$ and its equatorial radius is $6370\ \text{km}$. The diagram is shown in the figure.

1. a) Calculate the linear speed of a point on the equator, in $\text{km/h}$. Take $\pi = 3.14$ and round to the nearest whole number.
2. b) If this speed is so high, why do we not feel it? Give reasons for your answer.

A wheel rotates with an angular velocity of $\omega = 2\ \text{rad/s}$. Consider two points: one at $0.2\ \text{m}$ from the center and another at $0.4\ \text{m}$. The diagram is shown in the figure.

1. a) Calculate the linear speed of each point using $v = \omega \times r$.
2. b) Explain why both points have the same angular velocity but different linear speeds.

A drone's propeller has a radius of $0.15\ \text{m}$ and rotates at a frequency of $100\ \text{Hz}$.

1. a) What is its period?
2. b) Calculate the linear speed of the tip of the propeller. Take $\pi = 3.14$.

On the "flying chairs" ride, the chairs rotate at $6\ \text{m}$ from the center with a period of $12\ \text{s}$.

1. a) Calculate the linear speed of a chair. Take $\pi = 3.14$.
2. b) How long do $5$ revolutions take?
3. c) What distance does a chair travel in those $5$ revolutions?

The second hand of a wall clock is $0.15\ \text{m}$ long.

1. a) Calculate the linear speed of its tip, in $\text{m/s}$. Take $\pi = 3.14$ and round to three decimal places.
2. b) Express the result in $\text{cm/s}$, to one decimal place.
3. c) An ant walks at $2\ \text{cm/s}$. Which is faster, the ant or the tip of the second hand? Justify your answer using calculations.

Two friends get on a merry-go-round with a period of $8\ \text{s}$: one at $2\ \text{m}$ from the center and the other at $5\ \text{m}$.

1. a) Calculate the linear speed of each person. Take $\pi = 3.14$ and round to one decimal place.
2. b) Calculate the ratio of the two speeds and compare it with the ratio of the radii.
3. c) Calculate the angular velocity of the merry-go-round using $\omega = \frac{2 \pi}{T}$ and explain why it is the same for both friends even though their linear speeds are different.

A cyclist rides at $6.28\ \text{m/s}$ on a bicycle whose wheels have a radius of $0.5\ \text{m}$.

1. a) Calculate the rotation frequency of the wheels. Take $\pi = 3.14$.
2. b) How many revolutions does each wheel make in $5\ \text{min}$ of pedalling?
3. c) What distance does the cyclist cover in that time? Check that it equals the number of revolutions multiplied by the circumference of the wheel.

A turntable rotates at $45\ \text{rpm}$. Compare a point on the surface of the record located at $0.05\ \text{m}$ from the center with another located at $0.15\ \text{m}$.

1. a) Calculate the angular velocity of the record in $\text{rad/s}$. Take $\pi = 3.14$.
2. b) Calculate the linear speed of each point, rounded to two decimal places.
3. c) Calculate the ratio of the linear speeds and of the radii, and state the conclusion: how does $v$ depend on $r$ when $\omega$ is fixed?

A microwave turntable has a radius of $0.15\ \text{m}$ and takes $10\ \text{s}$ to complete one revolution.

1. a) Calculate the linear speed of a point on the edge, in $\text{m/s}$, and then express it in $\text{cm/s}$. Take $\pi = 3.14$.
2. b) If a plate of food is heated for $2.5\ \text{min}$, how many revolutions does it make?
3. c) Explain the purpose of having the turntable rotate while heating.

A large Ferris wheel with a radius of $60\ \text{m}$ takes $30\ \text{min}$ to complete one full revolution.

1. a) Calculate the linear speed of a capsule, in $\text{m/s}$. Take $\pi = 3.14$ and round to two decimal places.
2. b) A person walking travels at about $1.4\ \text{m/s}$. How many times faster is the person walking than the capsule?
3. c) In light of the result, explain why the passengers on the Ferris wheel hardly notice the movement.

In a clinical laboratory, a centrifuge spins sample tubes at $50\ \text{Hz}$ in a circle of radius $0.1\ \text{m}$.

1. a) What is the period of rotation?
2. b) Calculate the linear speed of a tube. Take $\pi = 3.14$.
3. c) How many revolutions does the centrifuge make in $2\ \text{min}$ of operation?

A ceiling fan rotates at a frequency of $4\ \text{Hz}$. Consider a point A halfway along the blade ($0.3\ \text{m}$ from the axle) and a point B at the tip ($0.6\ \text{m}$).

1. a) Calculate the linear speed of each point. Take $\pi = 3.14$ and round to one decimal place.
2. b) For safety reasons, the tip of the blade must not exceed $20\ \text{m/s}$. Calculate the maximum frequency at which the fan can rotate, rounded to one decimal place.
3. c) Justify, using the relationship between $v$, $\omega$ and $r$, why the critical point is the tip of the blade and not the center.