Torque & Acceleration (Rotational Dynamics) Practice Problems

You pull with your hand on a string wound around a cylinder of diameter 30 cm and weight 39 N. The cylinder is free to rotate around a horizontal axle passing through its center of mass. The 30 N applied force is directed toward the positive x-axis and pulls the string tangentially from the cylinder as shown in the image. Determine i) the magnitude and ii) the direction of the force that the axle exerts on the cylinder.

A box weighing 10 N rests on a horizontal table. A lightweight cord attached to the box passes through a cylindrical pulley with a mass of 1 kg and a diameter of 18 cm. A 6 N weight is suspended from the other end of the cord as shown in the image. After the system is released, determine the acceleration of the box.

A toy is designed so that when charged it stores rotational energy on a spherical ball of diameter 60 mm and mass 890 g distributed uniformly. The ball is pivoted using a frictionless axle passing through its center. A dent on the toy's body scratches the ball at a point on its center creating 0.0820 N friction force at the point of contact. Determine the angular acceleration of the ball.

A steam engine has its flywheel pivoted about a vertical axle through its center. The flywheel has a moment of inertia of 2200 kg•m², a radius of 1.80 m, and spins with insignificant friction. An engineer studying the flywheel applies a tangential 35.0 N force to the flywheel's circumference for 8.0 s. Determine the work done by the engineer on the flywheel it was at rest when the force was applied.

A string is wound around a pulley. The free end of the string is attached to a metallic block. The pulley has a mass of 8.50 kg, a radius of 220 mm, and behaves like a uniform cylinder. You release the block from rest and notice that the block unwinds a 17.0 m length of the string in 6.50 s. What is the string's tension?

A solid disk with a moment of inertia of 5.4kg•m² measured relative to its axis of rotation is required to spin at an angular speed of 850 rev/min. Determine the torque that will bring the disk to this speed within 4.50 s when the disk is initially at rest.

A jewelry box of mass 1 kg is attached to the end of a thin string. The string is wound around a solid, uniform disk of mass 6 kg and radius 10 cm. The disk rotates about a fixed axis that passes through its center. What is the speed of the box right before hitting the ground if the box is released from rest from a height of 1m above the ground?

A chocolate box of mass m₁ = 1 kg and an ice bucket of mass m₂ = 3 kg are connected by a rope over a pulley of radius 20 cm. The chocolate box rests on a frictionless horizontal table. The hanging ice bucket is released vertically from rest and reaches a speed of 2 m/s after 1 s. Let T₁ be the tension in the part of the cord attached to m₁ and T₂ be the tension in the part of the cord attached to m₂. Find the tension in each part of the rope.

A 450 g globe (solid sphere) of radius 12.5 cm initially motionless revolves about its central vertical axis under the influence of a constant tangential and horizontal force of magnitude 1.25 N. The force is applied along the line around the middle of the globe. The angular speed of the globe 15 s after the application of the force is 36 rpm. Assume that there is a constant frictional force f between the globe and the axis of rotation. Calculate the torque produced by f.

The revolving cylinder that transmits power to a spin coater has a radius of 2.0 cm and a mass of 75 g. One end of the cylinder is attached to the center of a circular plate. The plate has a mass of 600 g and a radius of 15 cm. During an experiment, the spin coater was revolving at a constant angular speed of 165 rpm. In an emergency, you needed to stop the disk. You shut off the electrical power and applied a tangential friction force (f) to the cylinder. The plate stopped in 8.0 s. Calculate the magnitude of f.

A 13 kg cylindrical flywheel in a spin bike is uniformly accelerated from rest to a rotational speed of 14 rev/s in 5.0 s. Given that the flywheel has a radius of 0.38 m and that the flywheel has a uniform mass distribution, determine what the value of torque applied to it will be.

A Buddhist monk is holding a hand-held prayer wheel. A small metal ball of 0.20 kg is attached to the head of the prayer wheel by a string of length 0.19 m. The monk starts to move the prayer wheel which makes the metal ball move in a horizontal circle, accelerating it to a rotational speed of 66 rpm in 5.5 s from rest. Determine the magnitude and origin of the torque on the small ball.

Suppose an arm utilizing the triceps muscle accelerates a 3.7-kg weight at 7.1 m/s². Calculate the force exerted by the triceps muscle while ignoring arm mass.

A small object attached to a thread is being swung around counterclockwise in a circle of radius R_o with uniform angular acceleration α. The position vector r of the object is given by: r̄ = î R_o cos Φ + ĵ R_o sin Φ where Φ = ω_it + (1/2)αt², here ω_i denotes the initial angular velocity and t is the time elapsed. Given that the mass of the object is m, and its moment of inertia is I, find its tangential acceleration a_t, and using τ̅ = Iα̅, find the torque acting on it.