1
IIT-JEE 2012 Paper 2 Offline
+3
-0.75
Two identical discs of same radius R are rotating about their axes in opposite directions with the same constant angular speed $$\omega$$. The discs are in the same horizontal plane. At time t = 0, the points P and Q are facing each other as shown in the figure. The relative speed between the two points P and Q is vr. In one time period (T) of rotation of the discs, vr as a function of time is best represented by
A
B
C
D
2
IIT-JEE 2012 Paper 2 Offline
+3
-0.75
Consider a disc rotating in the horizontal plane with a constant angular speed $$\omega$$ about its centre O. The disc has a shaded region on one side of the diameter and an unshaded region on the other side as shown in the figure. When the disc is in the orientation as shown, two pebbles P and Q are simultaneously projected at an angle towards R. The velocity of projection is in the y - z plane and is same for both pebbles with respect to the disc. Assume that (i) they land back on the disc before the disc has completed $${1 \over 8}$$ rotation, (ii) their range is less than half the disc radius, and (iii) $$\omega$$ remains constant throughout. Then

A
P lands in the shaded region and Q in the unshaded region.
B
P lands in the unshaded region and Q in the shaded region.
C
Both P and Q land in the unshaded region.
D
Both P and Q land in the shaded region.
3
IIT-JEE 2012 Paper 2 Offline
+3
-1

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 axis 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 centre of mass of the disc about the z-axis, and (ii) a rotation of the disc through an instantaneous vertical axis passing through its centre of mass (as it 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 systems as shown in the figure.

Case (a) : The disc with its face vertical and parallel to x-z axis;

Case (b) : The disc with its face making an angle of 45$$^\circ$$ with xy-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?

A
It is vertical for both Cases (a) and (b).
B
It is vertical for Case (a); and is at 45$$^\circ$$ to the xz-plane and lies in the plane of the disc for Case (b).
C
It is horizontal for Case (a); and is 45$$^\circ$$ to the xz-plane and is normal to the plane of the disc for Case (b).
D
It is vertical for Case (a); and is 45$$^\circ$$ to the xz-plane and is normal to the plane of the disc for Case (b).
4
IIT-JEE 2012 Paper 2 Offline
+3
-1

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 axis 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 centre of mass of the disc about the z-axis, and (ii) a rotation of the disc through an instantaneous vertical axis passing through its centre of mass (as it 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 systems as shown in the figure.

Case (a) : The disc with its face vertical and parallel to x-z axis;

Case (b) : The disc with its face making an angle of 45$$^\circ$$ with xy-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?

A
It is $$\sqrt2$$$$\omega$$ for both cases.
B
It is $$\omega$$ for case (a); and $$\omega$$ / $$\sqrt2$$ for case (b).
C
It is $$\omega$$ for case (a); and $$\sqrt2$$$$\omega$$ for case (b).
D
It is $$\omega$$ for both cases.
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