Directions: Each question contains STATEMENT-1 and STATEMENT-2. Choose the correct answer(ONLY ONE option is correct ) from the following-

Statement 1 : When you lean behind over the hind legs of the chair, the chair falls back after a certain angle.

Statement 2 : Centre of mass lying outside the system makes the system unstable.

(a)Statement -1 is false, Statement-2 is true
(b)Statement -1 is true, Statement-2 is true;Statement -2 is a correct explanation for Statement-1
(c)Statement -1 is true, Statement-2 is true; Statement -2 is not a correct explanation for Statement-1
(d)Statement -1 is true, Statement-2 is false

Solution

(d) Statement -1 is true, Statement-2 is false

A small disc of radius 2 cm is cut from a disc of radius 6 cm.If the distance between their centres is 3.2 cm, what is the shift in the centre of mass of the disc?

0.4 cm
2.4 cm
1.8 cm
1.2 cm

The radius of gyration of a body about an axis at a distance 6 cm from its centre of mass is 10 cm. Then, its radius of gyration about a parallel axis through its centre of mass will be

80 cm
8 cm
0.8 cm
80 m

Solution

From the theorem of parallel axes, the moment of inertia

I = I^CM + Ma²

where I_CM is moment of inertia about centre of mass and a is the distance of axis from centre.

mk² = m(k1)²+ m(6)²

(∵ I = mk² where k is radius of gyration)

or, k² = (k¹)² + 36

or,100 = (k¹)² + 36

⇒ (k¹)²= 64 cm

∴ k¹= 8 cm

Point masses 1, 2, 3 and 4 kg are lying at the points (0, 0, 0),(2, 0,0), (0, 3, 0) and (–2, –2, 0) respectively. The moment of inertia of this system about X-axis will be

43 kg –m²
34 kg–m²
27 kg – m²
72 kg – m²

Solution

Moment of inertia of the whole system about the axis of rotation will be equal to the sum of the moments of inertia of all the particles.

∴ I = I₁ + I₂ + I₃ + I₄

= 0+ 0 + 27 +16 = 43 kg m₄

A solid sphere and a hollow sphere of the same material and of same size can be distinguished without weighing

by determining their moments of inertia about their coaxial axes
by rolling them simultaneously on an inclined plane
by rotating them about a common axis of rotation
by applying equal torque on them

Solution

Acceleration of solid sphere is more than that of hollow sphere, it rolls faster, and reaches the bottom of the inclined plane earlier.Hence, solid sphere and hollow sphere can be distinguished by rolling them simultaneously on an inclined plane.

A solid cylinder of mass 20 kg rotates about its axis with angular speed 100 rad/s. The radius of the cylinder is 0.25 m. The K.E. associated with the rotation of the cylinder is

3025 J
3225 J
3250 J
3125 J

Solution

A boy is spinning on a rotating stool with his arms down to his sides. Suddenly he stretches his arms. Which of the following is true regarding his angular speed (ω) and the angular momentum (L) ?

ω decreases but L remains same
both ω and L decrease
ω remains constant but L decreases
ω increases but L remains constant

Solution

I₁ω₁= I₂ω₂

As I₂ > I₁,

∴ ω₂ < ω₁

A solid cylinder of mass 2 kg and radius 0.1 m rolls down an inclined plane of height 3m without slipping. Its rotational kinetic energy when it reaches the foot of the plane would be :

22.7 J
19.6 J
10.2 J
9.8 J

Solution

If the angular velocity of a body rotating about an axis is doubled and its moment of inertia halved, the rotational kinetic energy will change by a factor of :

4
2
1
¹⁄₂

Solution

K = ¹⁄₂ Iω²

K' = ¹⁄₂(¹⁄₂)(2ω)² = 2K

One quarter sector is cut from a uniform circular disc of radius R. This sector has mass M. It is made to rotate about a line perpendicular to its plane and passing through the centre of the original disc. Its moment of inertia about the axis of rotation is

¹⁄₂ mR²
¹⁄₄ mR²
¹⁄₈ mR²
√2 mR²

Solution

For complete disc with mass '4M', M.I. about given axis = (4M)(R²/2) = 2 MR²

Hence, by symmetry, for the given quarter of the disc

M.I. = 2 MR²/4 = ¹⁄₂ MR²

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