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प्रश्न
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?
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उत्तर
Let α be the angular acceleration produced in the sphere.
Rotational equation of motion,
\[F \times R - f_r \times R = I\alpha\]
\[\Rightarrow F = \frac{2}{5}mR\alpha + \mu mg........(1)\]
Translational equation of motion,
\[F = ma - \mu mg\]
\[ \Rightarrow a = \frac{\left( F + \mu mg \right)}{m}\]
For pure rolling, we have
\[\alpha = \frac{a}{R}\]
\[\Rightarrow \alpha = \frac{\left( F + \mu mg \right)}{mR}\]
Putting the value of \[\alpha\] in equation (1), we get
\[F = \frac{2}{5}\frac{mR\left( F + \mu mg \right)}{mR} + \mu mg\]
\[ \Rightarrow F = \frac{2}{5}\left( F + \mu mg \right) \mu mg\]
\[ \Rightarrow F = \frac{2}{5}F + \left( \frac{2}{5} \times \frac{2}{7} \times 0 . 5 \times 10 \right) + \left( \frac{2}{7} \times 0 . 5 \times 10 \right)\]
\[ \Rightarrow \frac{3F}{5} = \frac{4}{7} + \frac{10}{7} = 2\]
\[ \Rightarrow F = \frac{5 \times 2}{3} = \frac{10}{3} = 3 . 3 N\]
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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.
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;
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 _______.
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 sphere cannot roll on
Discuss the interlink between translational, rotational and total kinetic energies of a rigid object rolls without slipping.
Answer in Brief:
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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 ______.
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(Force constant of the spring = 36 N/m)
When a sphere rolls without slipping, the ratio of its kinetic energy of translation to its total kinetic energy is ______.
