###### Advertisements

###### Advertisements

Derive an expression for kinetic energy, when a rigid body is rolling on a horizontal surface without slipping. Hence find kinetic energy for a solid sphere.

###### Advertisements

#### Solution

Total Rolling kinetic energy = Translational K. E. + Rotational K. E.

`=1/2 MV^2+1/2 Iomega^2`

But

`omega=V/R `

`"Total Rolling K.E" = 1/2 MV^2+1/2 I(V^2/R^2)`

`"for solid sphere" I=2/5 MR^2`

`"Total Rolling K.E."=1/2 MV^2+1/2 (2MR^2)/5(V^2/R^2)`

`=1/2 MV^2+1/5 MV^2`

`=7/10 MV^2`

#### APPEARS IN

#### RELATED QUESTIONS

Read each statement below carefully, and state, with reasons, if it is true or false;

The instantaneous speed of the point of contact during rolling is zero.

Read each statement below carefully, and state, with reasons, if it is true or false;

A wheel moving down a perfectly *frictionless *inclined plane will undergo slipping (not rolling) motion

A solid sphere of mass 1 kg rolls on a table with linear speed 2 m/s, find its total kinetic energy.

If a rigid body of radius ‘R’ starts from rest and rolls down an inclined plane of inclination

‘θ’ then linear acceleration of body rolling down the plane is _______.

A stone of mass 2 kg is whirled in a horizontal circle attached at the end of 1.5m long string. If the string makes an angle of 30° with vertical, compute its period. (g = 9.8 m/s^{2})

Can an object be in pure translation as well as in pure rotation?

Two uniform solid spheres having unequal masses and unequal radii are released from rest from the same height on a rough incline. If the spheres roll without slipping, ___________ .

A hollow sphere and a solid sphere having same mss and same radii are rolled down a rough inclined plane.

A sphere cannot roll on

In rear-wheel drive cars, the engine rotates the rear wheels and the front wheels rotate only because the car moves. If such a car accelerates on a horizontal road the friction

(a) on the rear wheels is in the forward direction

(b) on the front wheels is in the backward direction

(c) on the rear wheels has larger magnitude than the friction on the front wheels

(d) on the car is in the backward direction.

A sphere can roll on a surface inclined at an angle θ if the friction coefficient is more than \[\frac{2}{7}g \tan\theta.\] Suppose the friction coefficient is \[\frac{1}{7}g\ tan\theta.\] If a sphere is released from rest on the incline, _____________ .

The following figure shows a smooth inclined plane fixed in a car accelerating on a horizontal road. The angle of incline θ is related to the acceleration a of the car as a = g tanθ. If the sphere is set in pure rolling on the incline, _____________.

A cylinder rolls on a horizontal place surface. If the speed of the centre is 25 m/s, what is the speed of the highest point?

A string is wrapped over the edge of a uniform disc and the free end is fixed with the ceiling. The disc moves down, unwinding the string. Find the downward acceleration of the disc.

A hollow sphere is released from the top of an inclined plane of inclination θ. (a) What should be the minimum coefficient of friction between the sphere and the plane to prevent sliding? (b) Find the kinetic energy of the ball as it moves down a length l on the incline if the friction coefficient is half the value calculated in part (a).

Discuss the interlink between translational, rotational and total kinetic energies of a rigid object rolls without slipping.

**Answer in Brief:**

A rigid object is rolling down an inclined plane derive the expression for the acceleration along the track and the speed after falling through a certain vertical distance.

A pendulum consisting of a massless string of length 20 cm and a tiny bob of mass 100 g is set up as a conical pendulum. Its bob now performs 75 rpm. Calculate kinetic energy and increase in the gravitational potential energy of the bob. (Use π^{2} = 10)

A disc of the moment of inertia I_{a} is rotating in a horizontal plane about its symmetry axis with a constant angular speed ω. Another disc initially at rest of moment of inertia I_{b} is dropped coaxially onto the rotating disc. Then, both the discs rotate with the same constant angular speed. The loss of kinetic energy due to friction in this process is, ______

The speed of a solid sphere after rolling down from rest without sliding on an inclined plane of vertical height h is, ______

What is the condition for pure rolling?

What is the difference between sliding and slipping?

Discuss rolling on an inclined plane and arrive at the expression for acceleration.

A uniform disc of mass 100g has a diameter of 10 cm. Calculate the total energy of the disc when rolling along with a horizontal table with a velocity of 20 cms^{-1}. (take the surface of the table as reference)

A solid sphere rolls down from top of inclined plane, 7m high, without slipping. Its linear speed at the foot of plane is ______. (g = 10 m/s^{2})

A solid sphere of mass 1 kg and radius 10 cm rolls without slipping on a horizontal surface, with velocity of 10 emfs. The total kinetic energy of sphere is ______.

A man is supported on a frictionless horizontal surface. It is attached to a string and rotates about a fixed centre at an angular velocity `omega`. The tension in the strings is F. If the length of string and angular velocity are doubled, the tension in string is now ____________.

The power (P) is supplied to rotating body having moment of inertia 'I' and angular acceleration 'α'. Its instantaneous angular velocity is ______.

A ring and a disc roll on horizontal surface without slipping with same linear velocity. If both have same mass and total kinetic energy of the ring is 4 J then total kinetic energy of the disc is ______.

A solid sphere is rolling on a frictionless surface with translational velocity 'V'. It climbs the inclined plane from 'A' to 'B' and then moves away from Bon the smooth horizontal surface. The value of 'V' should be ______.

The angular velocity of minute hand of a clock in degree per second is ______.

A 1000 kg car has four 10 kg wheels. When the car is moving, fraction of total K.E. of the car due to rotation of the wheels about their axles is nearly (Assume wheels be uniform disc)

An object is rolling without slipping on a horizontal surface and its rotational kinetic energy is two-thirds of translational kinetic energy. The body is ______.

A solid spherical ball rolls on an inclined plane without slipping. The ratio of rotational energy and total energy is ______.

A uniform disc of radius R, is resting on a table on its rim.The coefficient of friction between disc and table is µ (Figure). Now the disc is pulled with a force F as shown in the figure. What is the maximum value of F for which the disc rolls without slipping?

A circular disc reaches from top to bottom of an inclined plane of length 'L'. When it slips down the plane, it takes time ' t_{1}'. when it rolls down the plane, it takes time t_{2}. The value of `t_2/t_1` is `sqrt(3/x)`. The value of x will be ______.

The least coefficient of friction for an inclined plane inclined at angle α with horizontal in order that a solid cylinder will roll down without slipping is ______.

If x = at + bt^{2}, where x is the distance travelled by the body in kilometers while t is the time in seconds, then the unit of b is ______.

A solid sphere of mass 2 kg is rolling on a frictionless horizontal surface with velocity 6m/s. It collides on the free end of an ideal spring whose other end is fixed. The maximum compression produced in the spring will be ______.

(Force constant of the spring = 36 N/m)

The kinetic energy and angular momentum of a body rotating with constant angular velocity are E and L. What does `(2E)/L` represent?

The angular displacement of a particle in 6 sec on a circle with angular velocity `pi/3` rad/sec is ______.

When a sphere rolls without slipping, the ratio of its kinetic energy of translation to its total kinetic energy is ______.

A disc of mass 4 kg rolls on a horizontal surface. If its linear speed is 3 m/ s, what is its total kinetic energy?