Advertisements
Advertisements
प्रश्न
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).
Advertisements
उत्तर
It is given that a hollow sphere is released from the top of an inclined plane of inclination θ.
(a) To prevent sliding, the body will make only perfect rolling. In this condition, we have
\[mgl \sin \theta - f = ma............(1)\]
\[f \times R = \left( \frac{2}{3} \right) m R^2 \times \left( \frac{a}{R} \right)\]
\[ \Rightarrow f = \frac{2}{3} ma...........(2)\]
On putting this value in the equation (1), we get
\[mg \sin \theta - \frac{2}{3} ma = ma\]
\[\Rightarrow a = \frac{3}{5} g \sin \theta\]
From equation (1), we have
\[mg \sin \theta - f = \frac{3}{5} mg \sin \theta\]
\[ \Rightarrow f = \frac{2}{5} mg \sin \theta\]
\[ \Rightarrow \mu mg \cos \theta = \frac{2}{5} mg \sin\theta\]
\[ \Rightarrow \mu = \frac{2}{5} \tan \theta\]
(b)
\[\left( \frac{1}{5} \right) \tan \theta \left( mg \cos \theta \right) R = \frac{2}{3} m R^2 \alpha\]
\[ \Rightarrow \alpha = \frac{3}{10} \left( \frac{g \sin \theta}{R} \right)\]
\[a_c = g \sin \theta - \left( \frac{g}{5} \right) \sin \theta\]
\[ = \left( \frac{4}{5} \right) g \sin \theta\]
\[ \Rightarrow t^2 = \frac{2l}{a_c}\]
\[= 2l \left( 4g\frac{\sin \theta}{5} \right) \left( \frac{5}{2g \sin \theta} \right)\]
\[\therefore \omega = at \]
\[ K . E . = \frac{1}{2} m \nu^2 + \frac{1}{2} I \omega^2 \]
\[ = \frac{1}{2} m \left( 2al \right) + \frac{1}{2} l \left( a^2 t^2 \right)\]
\[ = \frac{1}{2} m \left( 4g \frac{\sin \theta}{5} \right) \times 2 \times l + \frac{1}{2} \times \frac{2}{3} m R^2 \times \frac{9}{100}\]
\[ = \left( \frac{\sin^2 \theta}{R} \right) \times \left( \frac{5L}{2g \sin \theta} \right)\]
\[ = 4 mgl \frac{\sin \theta}{5} + 3 mgl \frac{\sin \theta}{40}\]
\[ = \frac{7}{8} mgl \sin \theta\]
APPEARS IN
संबंधित प्रश्न
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.
A solid sphere rolls down two different inclined planes of the same heights but different angles of inclination. (a) Will it reach the bottom with the same speed in each case? (b) Will it take longer to roll down one plane than the other? (c) If so, which one and why?
Read each statement below carefully, and state, with reasons, if it is true or false;
For perfect rolling motion, work done against friction is zero.
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/s2)
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
A solid sphere of mass 0⋅50 kg is kept on a horizontal surface. The coefficient of static friction between the surfaces in contact is 2/7. What maximum force can be applied at the highest point in the horizontal direction so that the sphere does not slip on the surface?
A disc of the moment of inertia Ia 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 Ib 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 difference between sliding and slipping?
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 ____________.
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 circular disc reaches from top to bottom of an inclined plane of length 'L'. When it slips down the plane, it takes time ' t1'. when it rolls down the plane, it takes time t2. The value of `t_2/t_1` is `sqrt(3/x)`. The value of x will be ______.
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 ______.
