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The general motion of a rigid body can be considered to be a combination of (i) a motion of its centre of mass about an axis, and (ii) its motion about an instantaneous exis passing through the centre of mass. These axes need not be stationary. Consider, for example, a thin uniform disc welded (rigidly fixed) horizontally at its rim to a massless, stick as shown in the figure. When the disc-stick system is rotated about the origin on a horizontal frictionless plane with angular speed omega the motion at any instant can be taken as a combination of (i) a rotation of the disc through an instantaneous vertical axis passing through its centre of mass (as is seen from the changed orientation of points P and Q). Both these motions have the same angular speed omega in this case Now consider two similar system as shown in the figure: Case (a) the disc with its face vertical and parallel to x-z plane, Case (b) the disc with its face making an angle of 45^@ with x-y plane and its horizontal diameter parallel to x-axis. In both the cases, the disc is welded at point P, and the systems are rotated with constant angular speed omega about the z-axis. Which of the following statements regarding the angular speed about the instantaneous axis (passing through the centre of mass) is correct?

The general motion of a rigid body can be considered to be a combination of (i) a motion of its centre of mass about an axis, and (ii) its motion about an instantaneous exis passing through the centre of mass. These axes need not be stationary. Consider, for example, a thin uniform disc welded (rigidly fixed) horizontally at its rim to a massless, stick as shown in the figure. When the disc-stick system is rotated about the origin on a horizontal frictionless plane with angular speed omega the motion at any instant can be taken as a combination of (i) a rotation of the disc through an instantaneous vertical axis passing through its centre of mass (as is seen from the changed orientation of points P and Q). Both these motions have the same angular speed omega in this case Now consider two similar system as shown in the figure: Case (a) the disc with its face vertical and parallel to x-z plane, Case (b) the disc with its face making an angle of 45^@ with x-y plane and its horizontal diameter parallel to x-axis. In both the cases, the disc is welded at point P, and the systems are rotated with constant angular speed omega about the z-axis. . Which of the following statements about the instantaneous axis (passing through the centre of mass) is correct?

List-I shows some arrangements in which motion of masses are described and list-II defines motion of centre of mass of the system (m + M). Match appropriate possible options in list-II

A body of mass m , radius R and moment of inertia I (about an axis passing through the centre of mass and perpendicular to plane of motion) is released from rest over a sufficiently rough ground (to provide accelerated pure rolling) find linear acceleration of the body.

In physical pendulum, the time period for small oscillation is given by T=2pisqrt((I)/(Mgd)) where I is the moment of inertial of the body about an axis passing through a pivoted point O and perpendicular to the plane of oscillation and d is the separation point between centre of gravity and the pivoted point. In the physical pendulum a speacial point exists where if we concentrate the entire mass of body, the resulting simple pendulum (w.r.t. pivot point O) will have the same time period as that of physical pendulum This point is termed centre of oscillation. T=2pisqrt((I)/(Mgd))=2pisqrt((L)/(g)) Moreover, this point possesses two other important remarkable properties: Property I: Time period of physical pendulum about the centre of oscillation (if it would be pivoted) is same as in the original case. Property II: If an impulse is applied at the centre of oscillatioin in the plane of oscillation, the effect of this impulse at pivoted point is zero. Because of this property, this point is also known as the centre of percussion. From the given information answer the following question: Q. A uniform rod of mass M and length L is pivoted about point O as shown in Figgt It is slightly rotated from its mean position so that it performs angular simple harmonic motion. For this physical pendulum, determine the time period oscillation.

The pulley shown in figure has a moment of inertias I about its xis and mss m. find the tikme period of vertical oscillastion of its centre of mass. The spring has spring constant k and the string does not slip over the pulley.

In physical pendulum, the time period for small oscillation is given by T=2pisqrt((I)/(Mgd)) where I is the moment of inertial of the body about an axis passing through a pivoted point O and perpendicular to the plane of oscillation and d is the separation point between centre of gravity and the pivoted point. In the physical pendulum a speacial point exists where if we concentrate the entire mass of body, the resulting simple pendulum (w.r.t. pivot point O) will have the same time period as that of physical pendulum This point is termed centre of oscillation. T=2pisqrt((I)/(Mgd))=2pisqrt((L)/(g)) Moreover, this point possesses two other important remarkable properties: Property I: Time period of physical pendulum about the centre of oscillation (if it would be pivoted) is same as in the original case. Property II: If an impulse is applied at the centre of oscillatioin in the plane of oscillation, the effect of this impulse at pivoted point is zero. Because of this property, this point is also known as the centre of percussion. From the given information answer the following question: Q. For the above question, locate the centre of oscillation.