Uniform Circular Motion Practice Problems

A 75-cm-long, massless, horizontal bar is connected at one end to an electric motor that rotates the bar at a constant angular speed of 48 rpm. A 250-gram ball is attached to the other end. Calculate the magnitude of the ball's velocity.

A weather satellite is in geostationary orbit, 36000 km above the equator. The satellite rotates around the earth at the same rate as the earth does. The satellite monitors the weather evolution of the same region of the earth using ultrasensitive detectors. Calculate the speed and acceleration of the satellite. The Earth's equatorial radius is 6378 km.

An extrasolar planet orbits around its star in a uniform circular motion of radius 1.43 × 10³ km. The planet completes one orbit in 21.3 days. What is the magnitude of the planet's angular velocity?

A meteorological satellite moves around our planet in a circular, uniform motion with a radius of 7242 km. Its orbital period is 101 min. Calculate the speed of the satellite.

A ceiling fan has a radius of 60 cm and 1200 rpm at full speed. A particle of mass 1 g sticks at the outer edge of the fan blade. Calculate the radial acceleration experienced by the particle when the fan runs at full speed.

A wind turbine contains 120 ft long blades and a propeller that rotates at 10 rpm. Calculate the linear speed at the blade tips, in ft/min.

The radius of Mars is 3,389.5 km and rotates once every 24.6 hours. The radial acceleration at the equator must be larger than g for items to leave the surface of the planet and enter space. What would be the needed rotational period on Mars for this to happen? (g on the surface of mars is 3.7 m/s²)

The radius of the earth is 6371 km, and a 200-kilogram satellite is launched into an orbit 700 km above the planet's surface. The satellite takes 1.64 hours to complete one orbit. Assuming a circular orbit, what is the radial acceleration of the satellite?

Centrifuges conveniently create any desired value of acceleration using rotation. Scientists use centrifuges to study the effects of high acceleration on human beings. However, large accelerations are not so nice to human beings. A 1.65 m tall person is strapped onto a 7.85 long centrifuge arm with their head at the outermost end. Calculate the difference in acceleration experienced by the head and feet of the person if they are subjected to a maximum acceleration of 9g.

Scientists use centrifuges to study the effects of large accelerations on human beings. The centrifuges are a convenient tool because of their ability to produce large accelerations. In a typical experiment, a person with a height of 1.58 m is centrifuged using a centrifuge arm that is 6.83 m long, with the person's head located at the outermost end of the centrifuge arm. Calculate the speed of the person's head if they are subjected to a maximum acceleration of 9g.

A plane flies in a completely horizontal circle at a constant speed of 300 km/h. Calculate the diameter of the circle if the plane's radial acceleration is equal to twice the gravitational acceleration.

A science teacher rotates a ball of mass 300 g attached to the end of a light cord in a horizontal plane at a steady angular speed of 18 rpm. The ball moves in a circle of diameter 2.0 m.  Find the tension in the cord.

A 25.0-kg object is attached to one end of a light, rigid cord. The object rotates in a horizontal circle of radius 5.0 m at a constant angular speed. The object completes one rotation in 8.5 seconds. Find the tension in the cord.

Two tiny spheres of mass 100 g are attached to two light sticks of length 75 cm, as shown in the figure. The system is mounted on a rotor. When the rotor turns at a constant angular velocity of 22 rpm, the spheres move along a horizontal circle. Find the tension in the stick attached to the rotor.

Two aircraft carriers are located on a line around the middle of the earth, halfway between the North and South poles. A distance of 1250 km separates the two aircraft carriers. The travel time for a cargo aircraft between the two carriers is 3.5 hours. Calculate the angular velocity of the cargo aircraft, in degrees per hour, with respect to the earth's surface. The distance from the earth's center to a point located on the line around the middle of the planet and on its surface is 6378 km.

The latitude of The Sydney Opera House is 33.85°, south. Calculate the linear speed of the Sydney Opera House. The mean radius of the earth is 6371 km.

An eagle dives into a pond to catch a fish. Just before it reaches the pond, it pulls out of the dive by moving in an arc of radius 0.0200 km. If the speed of the eagle was 157 km/h (43.6 m/s), calculate its acceleration in g's.

Consider a Ferris wheel rotating in the vertical plane, with the position of a cabin modeled as a point moving in a circular path. The position in the xy-plane is given by $\vec{r}=(15.0\mathrm{\ m})\cos [(0.500\ \mathrm{rad}/\mathrm{s})t]\hat{i}+(15.0\mathrm{\ m})\sin [(0.5\mathrm{00\ rad}/\mathrm{s})t]\hat{j}$ , where r is in meters, and t is in seconds. Calculate the velocity and acceleration of the cabin as functions of time.

Consider a horse on a merry-go-round moving along a circular path in the vertical plane. The position is given by $\vec{r}=(5.0 \mathrm{~m}) \cos [(2.0 \mathrm{rad} / \mathrm{s}) t] \hat{i}+(5.0 \mathrm{~m}) \sin [(2.0 \mathrm{rad} / \mathrm{s}) t] \hat{j}$, where r is in meters, and t is in seconds. Determine the horse's acceleration vector.

A low-flying jet plane moves in a vertical circular path such that it is right side up at the topmost point of its trajectory. If the diameter of the circular path is 32 m, how many revolutions per minute must it make for the pilot of the jet to feel weightless at the topmost point of the trajectory?

A small ball (mass m₁) is rotating horizontally in a circle on a frictionless table, connected to a hanging mass (mass m₂) by a light string passing through a hole in the table's center, as depicted in the diagram below. Determine the speed of the ball in terms of m₁, m₂, g (acceleration due to gravity ) and r (radius of the circular trajectory).

Imagine a small ball positioned within a rotating drum, which is open at the top. The drum rotates about a vertical axis with a frequency f, while the ball is restricted to traverse the inner frictionless circular path of radius R. Calculate the angle $\phi$ at which the ball reaches equilibrium inside the drum, signifying that it does not ascend or descend along the circular trajectory.

A car moving steadily takes a curved path with a radius of 218 m. An air freshener hanging from the ceiling tilts to an angle of 19.5°. Calculate the car's velocity.

An adventurer intends to leap across a chasm using a rope tied to an overhanging branch. If she can exert a maximum force of 1360 N on the rope, determine the highest speed she can manage at the bottom of her leap. She weighs 79 kg, and the rope's length is 4.9 m.

You rotate a metal ball attached to a 0.48-m chain directly beneath you, creating a circular motion on the ground. The ball completes one full rotation every 0.68 s. Determine the angle that the chain forms with the vertical.

Inside a vertical circular track shown in the diagram below, a marble with mass (m) rolls without friction. The track rotates about a vertical axis with a frequency (f). If the frequency (f) is 3.00 revolutions per second and the radius (r) is 20.0 cm, what is the angle θ?

In the diagram provided below, a small ball with mass (m) is confined to move within a vertical cylindrical disc of radius (r), which rotates about a vertical axis at a frequency (f). Is it feasible for the ball to reach the height of the center of the circle (θ = 90°)? Consider the inner surface of the disc is frictionless.

A washing machine operates its spin cycle to remove water from clothes. The drum of the washing machine has a radius of 0.30 meters and rotates at a frequency of 1600 revolutions per minute (rpm). Determine the minimum coefficient of static friction between the clothes and the inner surface of the drum required to prevent the clothes from slipping down as the drum spins.

Suppose a marker is placed 12.0 cm from the axis of a spinning platform whose speed can be varied. Determine the static friction coefficient between the marker and the spinning platform. Assume that when the platform's speed increases, the marker remains stationary until it reaches 39.0 rpm, at which point it slides off.

An insect crawls outward from the center of a spinning record player, which turns at 66 rpm. Given a 0.45 static friction coefficient between the insect and the record player, how far does the insect move before slipping? How does an insect behave after moving farther out?