Centripetal Forces Practice Problems

A 1.0 kg stone is attached to one end of a rope, which has negligible mass. The stone moves in a uniform circular motion in a vertical circle of radius 2.0 m. Calculate the tension at the bottom point, if the tension in the rope is zero at the topmost point.

A 5.0 kg object is attached to one end of a rope, which has negligible mass. The object moves in a uniform circular motion in a vertical circle of radius 2.0 m. Calculate the period of revolution, if the tension in the rope is zero at the topmost point.

You are riding a roller coaster that has a 60 m-diameter circular loop with a top speed of 25 m/s. What is your apparent weight at the bottom of the loop if your mass is 50 kg? What is your apparent weight at the top of the loop if the roller coaster's speed has decreased to 20 m/s?

A 5.0 kg object is attached to one end of a rod which has negligible mass. The object moves in a uniform circular motion in a vertical circle of radius 2.0 m. If the period of revolution is 1.2 s, calculate the speed of the object while it is revolving.

A 55 kg skateboarder is moving down a ramp with a 5.0 m radius of curvature. If at the bottom of the ramp he reaches a speed of 8.0 m/s, what is the upward force acting on him when he is at the bottom of the ramp?

A 50 kg child is riding a ferris wheel with a 15 m radius and a period of 20 s. What is the normal force that the seat exerts on the child at the top of the ride?

A 70 kg person is riding a Ferris wheel with a radius of 40.0 m and a tangential speed of 6.0 m/s. Calculate the normal force exerted on the passenger at the bottom.

In an amusement park, a design of the carousel has two wires, one of which is horizontal, and the seat is attached to them. A horizontal circle of radius 7.50 m is formed by the seat as it swings at a speed of 20.0 rpm (rev/min). Determine the tension in each cable if the seat weighs 200 N and a person weighing 850 N is seated in it.

The merry-go-round at an amusement park is made up of a vertical center shaft with a number of horizontal arms attached to it at the top. The top end of a 7.00 m long cable is attached to an arm, 4.00 m from the central shaft and suspends a seat from the lower end. If the cable holding a seat makes a 60.0° angle with the vertical, find the duration of one swing revolution.

On a smooth floor, a ballet dancer weighing 50 kg is spinning at 3.0 rotations per second while holding outstretched arms. 1.80 meters is the distance from one hand to the other. The average hand weighs about 1.50 percent of the body, per biometric studies. How much force must be applied horizontally by her wrist on her hand?

A paper clip of mass 2 g is placed 6 cm from the axis of rotation of a spin coater. The static and kinetic friction coefficient between the paperclip and the spin coater surface is mu(s) =0.95 and mu(k) = 0.64, respectively. The spin coater starts from rest and accelerates until it reaches a speed of 20 rpm. Identify the maximum angular velocity of the paper clip. Will the paperclip remain on the spin coater, or will it slide off?

Ali wants to test the hypothesis that the weight of an object varies depending on its location. For this purpose, he measures the weight of a book (m = 10 kg) at the North Pole and the Equator. Assuming the Earth to be spherical, will the scale readings be the same or different?

A sphere of mass m is attached to a rope of length L/2. The sphere rotates in a circle at a constant speed. The rope is slanted downward at an angle θ with the horizontal. Derive an expression for the angular velocity ω of the sphere in terms of L and θ.

A musical toy consists of a wooden ball that rotates in a horizontal circle around a central axis. The mass of the ball is 0.3 kg and is supported by two wires as shown in the figure. Determine the speed at which the tension in both wires is the same if the ball revolves at a constant speed.

Consider a scenario in which a square metallic object of mass 1 kg hangs from a 3 m long rope. If the object is displaced from equilibrium and allowed to oscillate, moving toward the bottom, when it reaches a speed of 3.2 m/s, it makes a 25° angle with the vertical. Determine the tension in the rope at this particular instant.

A paint-filled bucket with a tiny hole at the bottom is suspended from a rope. Suppose the bucket is initially at rest, but then it is given a disturbance that sets it in a circular motion (conical pendulum). The hole allows the paint to gradually drip out of the bucket, while the rope traces a cone's surface. Derive an expression for the angular velocity ω of the bucket in terms of L and r where L is the length of the rope suspending the paint-filled bucket, and r is the radius of the circle traced out by the bucket as it moves in a circular motion.

A child is playing with a yo-yo, which has a mass of 0.5 kg and is attached to a string that is 60 cm long. The child has set the yo-yo into motion by swinging it in a vertical circle. When the yo-yo is at an angle of 45° from the highest point on the circle, the tension in the string is measured to be 25 N. What is the speed of the yo-yo at this instant? Assume that there is negligible air resistance and no loss of energy due to friction.

A tech company designs a new rotational governor for The Great Inventor's Fair. The governor consists of a mass of 25 g attached to a spring with a spring constant of 12 N/m and relaxed length of 0.5R, where R is 10 cm. Using the spring force F_sp = k(r-L), where k is the spring constant measured in N/m, and L is the relaxed length of the spring, determine the critical angular velocity in rpm. Ignore gravity.

Hint: Rotational governors are used to control the speed of an elevator and prevent it from moving too quickly or free-falling in the event of a malfunction. The emergency brake activates when the metal blocks (as shown in the figure) reach a distance r = R from the center of rotation.

A child rotates a 75-g cube of side 3.0 cm tied to a light cord. The cube moves clockwise in a vertical circle of radius 0.5 m. At the position where the cube is moving straight up, the cube has a speed of 4.8 m/s. Calculate the strength of the total force applied to the cube. The cube is affected by air drag, and the cube drag coefficient is 1.15.

A tiny sphere of mass 250 g is attached to one end of a light cord of length 0.35 m. The other end is attached to the shaft of an electrical motor. When the motor is turned on, the sphere rotates clockwise in a vertical circle. The center of the circle is 1.25 m above the ground. When the cord is aligned with the horizontal and the sphere's velocity is directed toward the positive vertical axis, the cord is snapped. The highest vertical position attained by the sphere is 2.75 m with respect to the ground. Calculate the tension in the cord just before it snapped.

Elisa and Priya have designed an L-shaped track using cardboard, and the test object is a small aluminum foil ball. The track has a diameter of 15 cm, and the mass of the ball is 30 g. If the ball is moving around the track at 25 rpm, ignoring rolling friction, determine the magnitude of the net force exerted by the track on the ball.

A container filled with a viscous fluid that rotates at a constant angular velocity will result in a fluid surface that follows a parabolic curve. Find the equation for this curve in terms of z,ω, and r, where z is the height of the fluid surface, ω is the angular velocity, and r is the distance from the center of rotation to the point on the fluid surface. Remember, the fluid particles on the surface experience only two forces - gravity and normal force exerted by the fluid underneath it.

A toy is made using a frictionless spherical shell with an inner radius of 35 cm and a tiny pearl of mass 8.0 g. The pearl lies on the inner shell, as shown in the figure. The shell can rotate about a fixed vertical axis at various angular speeds. When playing, a child remarks that the pearl moves from the bottom of the shell to a higher position only when the shell rotates at an angular speed greater than ω_a. The line that connects the center of the shell to the pearl forms an angle α with the vertical. Find ω_a in revolutions per minute.

In an experimental amusement park ride, riders in a rotating cylindrical chamber will experience a simulated gravitational force. If the riders aboard the ride feel a simulated gravity of 0.80 g, determine the amount of time that is needed to complete one revolution. Note that the diameter of the chamber is 25 m.

Consider a person seated on one of the platforms of a merry-go-round swing ride at an amusement park, which has a radius (r) and rotates with an angular velocity (ω). Due to the centrifugal force, the person leans outward at the top of the ride. Calculate the angle ϕ at which the person deviates from the vertical direction, considering g, r, and ω. Assume the ride rotates in a horizontal plane.

Two equally weighted dancers clasp their hands in a dance hall and rotate around a shared axis once every 2.6 s. Assume each dancer's arm length is 0.90 m and each weight is 56.0 kg. Find the force with which they hold each other.