If a body is spinning about an axis with constant angular velocity ώ, each point of the body is moving in a circular path and, therefore, has some K.E. to determine the total K.E. of a spinning body, we imagine it to be composed of tiny pieces of mass m1, m2,…… if a piece of mass mi is at a distance ri from the axis of rotation, as shown in fig. 5.14, it is moving in a circle with speed

rotational-kinetic-energy

 kinetic-energy-equation-01

The rotational K.E of the whole body is the sum of the kinetic energies of all the parts. So we have

kinetic-energy-equation-02 We at once recognize that the quantity within the brackets is the moment of inertia I of the body. Hence, rotational kinetic energy is given by

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rotational-collison-the clutch

Rotational collision – the clutch

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If rolling or spinning bodies are present in a system, their rotational kinetic energy must be included as part of the total kinetic energy. Rotational kinetic energy is put to practical use by fly wheels, which are essential parts of many engines. Fly wheel stores energy between the powers stokes of the pistons, so that the energy is distributed over the full revolution of the crankshaft and hence, the rotation remains smooth.

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As the sphere rolls to the bottom of the incline, its gravitational potential energy is changed to kinetic energy of rotation and translation.

Rotational Kinetic Energy of a Disc and a Hoop

From equation 5.19 the rotational kinetic energy of a disc is

 kinetic-energy-equation-04When both start moving down an inclined plane of height h, their motion consists of both rotational and translational motions (Fig. 5.15). If no energy is lost against friction, the total kinetic energy of the disc or hoop on reaching the bottom of the incline must be equal to its potential energy at the top.

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Fig- 5.15

kinetic-energy-equation-05

Example 5.5: A disc without slipping rolls down a hill of height 10.0 m. if the disc starts from rest at the top of the hill, what is its speed at the bottom?

Solution:  Using Eq. 5.23

kinetic-energy-equation-06

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As the wheel rolls, it has both rotational and translational kinetic energies.