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physics A-Level Physics
 
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Kinematics of Uniform Circular Motion

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Detailed solution steps are available at:
 
https://en.openprof.com/wb/chapter:kinematics_of_uniform_circular_motion/4248/


Example 1.

Two cylinders of different radii are attached to a common axis which is on the geometric axis of the cylinders. The moment of inertia of both cylinders together is . A rope is wound on the thinner cylinder with a radius of 4 cm and tied to a body of mass 2 kg, which hangs freely on the rope. We lower the weight so that it falls with acceleration . Calculate the acceleration of the fall.

    
Example 2.

Determine which the following statements are correct:

  • During uniform rotation, the speed of a body changes with time.

  • The frequency of rotation is higher at higher values of period.

  • The tangential speed is proportional to the radius of rotation.

  • The frequency of rotation of an object at the equator is greater than the frequency of rotation of an object closer to the pole.

  • The minute hand on the clock rotates with a one-hour period.

  • If the frequency is zero, it means that the rotating body is at rest.

  • A turntable rotates at a speed of 45 revolutions per minute. The speed at which the gramophone needle slides across the record is constant during music playback.

  • The resultant of the forces acting on a body during uniform rotation is the centripetal force.

  • The centripetal force acts perpendicular to the direction of velocity of the rotating body.

    
3.

Two cylinders of the same length made of the same solid material have radii and . The cylinders are attached to a common axis, which is at the same time the geometric axis of both cylinders. What is the moment of inertia of both cylinders if the moment of inertia of the thinner cylinder is ?

    
4.

A hollow cylinder with a moment of inertia of can rotate about a vertical symmetrical axis. It is at rest. How much constant torque must we apply on the cylinder for it to rotate 10 rpm in 20 s?

    
5.

A wooden hollow cylinder with a radius of 1 m can rotate around a horizontal geometric axis. The moment of inertia of the cylinder is . The cylinder is stationary at first. A bullet with a mass of 10 g approaches at a speed of 850 m / s and sinks at an angle into the circumference of the cylinder. At what angular velocity does the cylinder rotate?

    
6.

An iron cylinder has a radius of 10 cm and a height of 20 cm. What is the moment of inertia of the cylinder with respect to the geometric axis? The density of iron is .

    
7.

Two cylinders with the same radii of 0.2 m rest on the same axis of rotation and are initially spaced in the vertical plane. The lower cylinder weighs 2 kg and the upper 1 kg. Initially, only the lower cylinder rotates at an angular velocity of 12 rad/s. At some point, the upper cylinder falls on the lower one. The cylinders initially slip until they both begin to rotate at the same angular velocity.

  • What is the angular velocity of both cylinders after the coupling?

  • What is the angular impulse with which the lower cylinder acts on the upper during coupling?


    
8.

A small stone attached to a thin string circulates in a circular path with a radius of 80 cm. Its linear speed is . During rotation, the cord hits a rod of radius 5 mm at a distance of 50 cm from the axis of rotation. What is the linear velocity of the stone when the string is wrapped around the rod three times?

    

9.

A homogeneous cylinder weighing 50 kg rolls without friction from the top of a 20 m long stick inclined at an angle to the horizontal. How long does it take the cylinder to get to the end of the stick?

    
10.

A fixed pulley consists of a homogeneous cylinder with a mass of 2 kg and a radius of 0.1 m. A rope is wrapped around the pulley and two weights are attached to the ends of the rope. The mass of the first weight is 300 g and the second 100 g. The device is also called an Atwood machine. At what acceleration do the weights move?

    
11.

A homogeneous cylinder with a mass of 100 kg is pushed on a horizontal surface so that it accelerates uniformly. It covers a distance of 20 m in 10 s. What should be the traction force of the cylinder? Friction is neglected.

    
12.

A 1 m long thin rod stands on the ice surface. The lower end of the rod is mounted on a rotating axis. The rod is hit by a pointed projectile at the height of 0.75 m from the lower end at a speed of and bounces off it elastically at a speed of . The mass of the rod is 3 kg and that of the projectile is 20 g.


Rod on the ice surface



  • At what angular velocity does the rod begin to rotate about its axis?


  • What is its angular velocity if the axis of rotation is in the middle of the length of the rod?


    The moment of inertia of a rod rotating around its end is . If it rotates around an axis in the middle, then the moment of inertia is .

    
13.

On a large round plate weighing 300 kg stands a man weighing 60 kg. The plate can rotate around the vertical axis without friction. In the beginning, the plate and the man are at rest. Suddenly, the man begins to move around the perimeter of the plate counterclockwise and moves through the whole perimeter of the plate in 6 s. At what angular velocity does the plate rotate?

    
14.

The frequency of rotation increases from to in . What is the average angular acceleration?

    
15.

A body moves in a circle of radius with a linear acceleration of . What is the angular acceleration of the body?

    
16.

A wheel rotates at a frequency of when it starts to brake uniformly. It rotates for another .

  • What is the angular deceleration?

  • How many turns does it make before it stops?

    
17.

A cyclist rides at a constant speed. At some point, it starts to accelerate uniformly, the angular acceleration of the wheel is and the acceleration lasts for after which it then continues to ride at a new constant speed.

  • How much has the wheel's angular velocity changed?

  • How many times does the wheel turn during acceleration?

    
18.

A body moves at an angular velocity of . At some point, it starts accelerating with angular acceleration .

  • What is the angular velocity after of accelerating?

  • What is the angle described by the radius during that period?