Question

16. A body rolls without slipping, find its kinetic energy​

Answer 1

Answer:

Energy of a Rolling Object

INTRODUCTION

In this experiment, we will apply the Law of Conservation of Energy to objects rolling down a ramp. As an object rolls down the incline, its gravitational potential energy is converted into both translational and rotational kinetic energy. The translational kinetic energy is

( 1 )

KEtrans = (1/2)mv2

whereas the rotational kinetic energy is

( 2 )

KErot = (1/2)Iω2

In this last equation ω is the angular velocity in radians/sec, and I is the object's moment of inertia. For objects with simple circular symmetry (e.g. spheres and cylinders) about the rotational axis, I may be written in the form:

( 3 )

I = kmr2

where m is the mass of the object and r is its radius. The geometric factor k is a constant which depends on the shape of the object:

k = 2/5 = 0.4 for a uniform solid sphere,

k = 1/2 = 0.5 for a uniform disk or solid cylinder,

k = 1 for a hoop or hollow cylinder.

If the object rolls without slipping, then the object's linear velocity and angular speed are related by

v = rω.

Substituting equation 3 and the expression for v into equation 2, we obtain:

( 4 )

Figure 1

Consider a round object rolling down a ramp as in the illustration above. Assuming no loss of energy we may write the conservation of energy equation as:

total energy at top of ramp = total energy at bottom of ramp,

Egravitational = Etranslational + Erotational

or,

( 5 )

mgh = (1/2)mv2 + (1/2)kmv2.

We can determine v by analyzing the motion of the ball after it leaves the table. Recalling that the horizontal and vertical motion of a projectile may be treated independently we have,

( 6a )

x = vt

and

( 6b )

H = (1/2)gt2

where t is the time of flight, x is the horizontal range, and H is the vertical height of the ramp above the floor. These two equations (6a and 6b) can be combined, eliminating t, to obtain the following expression for the velocity in terms of x and H.

( 7 )

v2 = gx2 / 2H

Therefore, the energy of the rolling object can be analyzed entirely in terms of the measured values: m, h, H, x, and the acceleration due to gravity g