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
Prove the result that the velocity v of translation of a rolling body (like a ring, disc, cylinder or sphere) at the bottom of an inclined plane of a height h is given by `v^2 = (2gh)/((1+k^2"/"R^2))`.
Using dynamical consideration (i.e. by consideration of forces and torques). Note k is the radius of gyration of the body about its symmetry axis, and R is the radius of the body. The body starts from rest at the top of the plane.
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.
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
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 sphere cannot roll on
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?
What is the difference between sliding and slipping?
Discuss rolling on an inclined plane and arrive at the expression for acceleration.
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/s2)
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 ______.
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 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 ' 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 ______.
If x = at + bt2, where x is the distance travelled by the body in kilometers while t is the time in seconds, then the unit of b is ______.
An inclined plane makes an angle 30° with the horizontal. A solid sphere rolling down an inclined plane from rest without slipping has linear acceleration ______. (g = acceleration due gravity) (sin 30° = 0.5)
