13 exercises on uniform circular motion (with solutions)

clock as an example of uniform circular motion

On this page you will find a collection of exercises on uniform circular motion, which is one of the fundamental concepts of classical physics and manifests itself in numerous everyday contexts and technological applications. It describes the motion of an object along a circular trajectory with constant angular velocity.

We will address key concepts such as rotational frequency and period, angular velocity, and tangential velocity, in a wide range of situations and levels of difficulty, from the simplest to the most complex applications.

Each problem is accompanied by a detailed solution, showing the mathematical steps required to arrive at the final answer, helping you gain a deep understanding of uniform circular motion.
These are primarily aimed at high school, high school, and three-year students, but anyone interested in the subject can benefit!

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Exercises solved on uniform circular motion

On frequency, tangential velocity and centripetal acceleration in uniform circular motion

A particle moves by uniform circular motion on a circumference of radius $r = 40 \ cm$ . Knowing that it makes one revolution every $0,5 \ s$

Calculates the frequency of motion, tangential velocity and centripetal acceleration

[ $2 \ Hz$ ; $5.03 \ m/s$ ; $63.2 \ m/s^2$ ]

Solution

On angular and tangential velocity in uniform circular motion

Consider two people on the rotating circular platform, at constant speed, of a merry-go-round. The first is at a distance of $1,0 \ m$ from the center, the second at $2,5 \ m$ from the center. Knowing that the rotational frequency of the carousel is $0.2 \ Hz$ ,

Calculate the angular velocities of the two and their tangential velocities

[ $1.3 \ rad/s$ ; $1.3 \ m/s$ ; $3.2 \ m/s$ ]

Solution

On the travel time of a section of a curve.

In a middle-distance race, an athlete enters a turn with tangential speed $v = 7.00 \ m/s$ , then kept constant along the entire curvilinear stretch. The athlete is in the outermost lane, at $700 \ cm$ From the center of curvature.

Calculate what centripetal acceleration it must maintain in order not to go off the trajectory
Assuming that the curve is a quarter circle, then calculate the travel time

[ $7 \ m/s^2$ ; $1,57 \ s$ ]

Solution

On the swept angle in uniform circular motion

Consider the minute hand of a watch. If it moves 23 minutes,

What corner did he sweep in this time?

[ $138^\circ$ ]

Solution

On the calculation of tangential velocity from frequency in uniform circular motion

A truck mounts wheels with a diameter of $60 \ cm$ . During the run, the frequency of rotation there each wheel is $683$ RPM.

Calculate how fast the vehicle is traveling

[ $77 \ km/h$ ]

Solution

On the uniform circular motion of two balls

A ball is thrown in a circular circuit of radius $1,27 \ m$ with initial tangential velocity of $8.00 \ m/s$ . At the same time and from the same position, a second ball is thrown in the opposite direction with tangential velocity $4.00 \ m/s$ .

Calculate their respective angular velocities and after how long they collide in the circuit. Assume the motion is uniform in time.

[ $6.30 \ rad/s$ ; $3.15 \ rad/s$ ; $0,665 \ s$ ]

Solution

On uniform circular motion and centripetal acceleration

The drum of a washing machine of diameter $d = 32 \ cm$ rotates with constant angular velocity and performs $700$ revolutions per minute

Calculate the rotation period of the basket
Calculate the centripetal acceleration of a point that is at the outermost part of the basket

[ $T = 0.085 \ s$ ; $a_c = 870 \ m/s^2$ ]

Solution

On centripetal acceleration

A vinyl record rotates with constant tangential velocity. If the centripetal acceleration at the distance $r_1 = 6.5 \ cm$ from the center of rotation is equal to $a_1 = 8.7 \ m/s^2$

Calculate the centripetal acceleration at distance $r_2 = 3.4 \ cm$

[ $a_2 = 16.5 \ m/s^2$ ]

Solution

On two wheels in uniform circular motion

Two wheels are in contact as shown in the figure. Both are moving in uniform circular motion, and their tangential velocity $v$ Is identical. If the ratio of the frequencies is equal to $2,2$

Uniform circular motion of two wheels in contact

Calculate the relationship between the periods
Calculate the ratio of centripetal accelerations

[ $\frac{T_B}{T_A} = 0.45$ ; $\frac{a_B}{a_A} = 2.2$ ]

Solution

On calculating the number of revolutions per minute of a wheel

A bicycle travels at the speed $v = 42.0 \ km/h$ . If the diameter of the bicycle wheels is equal to $d = 26.0 \ cm$

Calculates the number of revolutions per minute $N$ that accomplishes each wheel of the bicycle

[ $N = 860 \ rpm$ ]

Solution

On calculating the number of revolutions per minute of a wheel

A marble of moves within a circular track of uniform motion sweeping an angle $\theta = 12.8 \ rad$ at $t = 2.3 \ s$

Calculate the time required to travel $\theta = 432^\circ$
Calculate the time required to complete 8 laps

[ $t = 1.3 \ s$ ; $t = 9.0 \ s$ ]

Solution

On the calculation of the distance Earth - Moon

The period of revolution of the Moon around the Earth is equal to $T = 27.0 \ days$ . Assume that the Moon completes a circular orbit with tangential velocity equal to $v = 1.02 \cdot 10^3 \ m/s$

Calculate the distance Earth - Moon

[ $r = 3.84 \cdot 10^8 \m$ ]

Solution

On the uniform circular motion of a ball

Consider the system shown in the figure in which a ball attached to a rope of length $L = 6.6 \ m$ moves by uniform circular motion at tangential velocity $v = 9.8 \ m/s$ . If the angle of inclination of the rope with respect to the center of rotation is $\theta = 45^\circ$

Uniform circular motion of a ball attached to a rope

Calculate the angular velocity $\omega$

[ $\omega = 2.1 \ rad/s$ ]

Solution

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