(A) The torsional rigidity of a shaft is given by $C = \frac{\eta J}{L}$, where $\eta$ is the modulus of rigidity, $J$ is the polar moment of inertia,

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(A) The torsional rigidity of a shaft is given by $C = \frac{\eta J}{L}$, where $\eta$ is the modulus of rigidity, $J$ is the polar moment of inertia, and $L$ is the length. For a given torque $\tau$, the angle of twist $\theta$ is $\theta = \frac{\tau L}{\eta J}$. For a hollow shaft, the material is distributed further from the axis, which increases the polar moment of inertia $J$ for the same mass. Since $J_{hollow} > J_{solid}$ for the same mass and length, the torque required to produce a specific twist $\theta$ is higher for the hollow shaft. Therefore, the hollow shaft is stronger in torsion, and the reason correctly explains the assertion.

Class 11 Physics System of Particles and Rotational Motion Mix Example - System of Particles and Rotational Motion

Medium

$Assertion :$ $A$ hollow shaft is found to be stronger than a solid shaft made of the same material and having the same mass per unit length.
$Reason :$ The torque required to produce a given twist in a hollow cylinder is greater than that required to twist a solid cylinder of the same length and material.

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A
If both Assertion and Reason are correct and the Reason is a correct explanation of the Assertion.
B
If both Assertion and Reason are correct but Reason is not a correct explanation of the Assertion.
C
If the Assertion is correct but Reason is incorrect.
D
If both the Assertion and Reason are incorrect.

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System of Particles and Rotational Motion

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Mix Example - System of Particles and Rotational Motion

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$A$ solid sphere of radius $R$ has a moment of inertia $I$ about its geometric axis. It is melted and recast into a disc of radius $r$ and thickness $t$. If the moment of inertia of this disc about an axis tangent to its edge and perpendicular to its plane is also $I$,find the value of $r$.

$A$ wagon of $200\, kg$ is moving on a smooth track with a velocity of $2\, m/s$. $A$ man of $80\, kg$ runs in the wagon with a velocity such that the speed of the centre of mass of the system is zero. Find the relative velocity of the man with respect to the wagon in $m/s$.

State whether the following statements are true or false:$(1)$ Torque produces angular velocity in an object.$(2)$ For the rotational motion of a rigid body,the linear variables of all its particles are the same.

$A$ hoop of radius $r$ and mass $m$ rotating with an angular velocity $\omega_0$ is placed on a rough horizontal surface. The initial velocity of the centre of the hoop is zero. What will be the velocity of the centre of the hoop when it ceases to slip?

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State whether the following statements are true or false:
$(1)$ Torque produces angular velocity in an object.
$(2)$ For the rotational motion of a rigid body,the linear variables of all its particles are the same.