1
IIT-JEE 2008 Paper 2 Offline
MCQ (Single Correct Answer)
+3
-1

The nuclear charge (Ze) is non-uniformly distributed within a nucleus of radius R. The charge density $$\rho(r)$$ [charge per unit volume] is dependent only on the radical distance r from the centre of the nucleus as shown in figure. The electric field is only along the radial direction.

For a = 0, the value of d (maximum value of $$\rho$$ as shown in the figure) is

A
$${{3Ze} \over {4\pi {R^3}}}$$
B
$${{3Ze} \over {\pi {R^3}}}$$
C
$${{4Ze} \over {3\pi {R^3}}}$$
D
$${{Ze} \over {3\pi {R^3}}}$$
2
IIT-JEE 2008 Paper 2 Offline
MCQ (Single Correct Answer)
+3
-1

The nuclear charge (Ze) is non-uniformly distributed within a nucleus of radius R. The charge density $$\rho(r)$$ [charge per unit volume] is dependent only on the radical distance r from the centre of the nucleus as shown in figure. The electric field is only along the radial direction.

The electric field within the nucleus is generally observed to be linearly dependent on r. This implies

A
$$a = 0$$
B
$$a = {R \over 2}$$
C
$$a = R$$
D
$$a = {{2R} \over 3}$$
3
IIT-JEE 2008 Paper 2 Offline
MCQ (Single Correct Answer)
+3
-1

A uniform thin cylindrical disk of mass M and radius R is attached to two identical massless springs of spring constant $$k$$ which are fixed to the wall as shown in the figure. The springs are attached to the axle of the disk symmetrically on either side at a distance d from its centre. The axle is massless and both the springs and the axle are in a horizontal plane. The unstretched length of each spring is L. The disk is initially at its equilibrium position with its centre of mass (CM) at a distance L from the wall. The disk rolls without slipping with velocity $${\overrightarrow V _0} = {V_0}\widehat i$$. the coefficient of friction is $$\mu$$.

The net external force acting on the disk when its centre of mass is at displacement x with respect to its equilibrium position is :

A
$$- kx$$
B
$$- 2kx$$
C
$$- {{2kx} \over 3}$$
D
$$- {{4kx} \over 3}$$
4
IIT-JEE 2008 Paper 2 Offline
MCQ (Single Correct Answer)
+3
-1

A uniform thin cylindrical disk of mass M and radius R is attached to two identical massless springs of spring constant $$k$$ which are fixed to the wall as shown in the figure. The springs are attached to the axle of the disk symmetrically on either side at a distance d from its centre. The axle is massless and both the springs and the axle are in a horizontal plane. The unstretched length of each spring is L. The disk is initially at its equilibrium position with its centre of mass (CM) at a distance L from the wall. The disk rolls without slipping with velocity $${\overrightarrow V _0} = {V_0}\widehat i$$. the coefficient of friction is $$\mu$$.

The centre of mass of the disk undergoes simple harmonic motion with angular frequency $$\omega$$ equal to:

A
$$\sqrt {{k \over M}}$$
B
$$\sqrt {{{2k} \over M}}$$
C
$$\sqrt {{{2k} \over {3M}}}$$
D
$$\sqrt {{{4k} \over {3M}}}$$
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