1
MCQ (Single Correct Answer)

JEE Main 2016 (Online) 10th April Morning Slot

A bottle has an opening of radius a and length b. A cork of length b and radius (a + $$\Delta $$a) where ($$\Delta $$a < < a) is compressed to fit into the opening completely (See figure). If the bulk modulus of cork is B and frictional coefficient between the bottle and cork is $$\mu $$ then the force needed to push the cork into the bottle is :
A
($$\pi $$ $$\mu $$ B b) $$\Delta $$a
B
(2$$\pi $$ $$\mu $$ B b) $$\Delta $$a
C
($$\pi $$ $$\mu $$ B b) a
D
(4$$\pi $$ $$\mu $$ B b) $$\Delta $$a

Explanation

Bulk modulus, B  =  $${{\Delta P} \over {{{\Delta V} \over V}}}$$

Vi  =  $$\pi $$ (a + $$\Delta $$a)2b

Vf  =  $$\pi $$a2b

$$ \therefore $$   $$\Delta $$V = 2$$\pi $$ab$$\Delta $$a

$$ \therefore $$   $${{{\Delta V} \over V}}$$ = $${{2\pi ab\Delta a} \over {\pi {a^2}b}}$$ = $${{2\Delta a} \over a}$$

$$ \therefore $$   $$\Delta $$P = B $$ \times $$ $${{{\Delta V} \over V}}$$

      = B $$ \times $$ $${{2\Delta a} \over a}$$

Normal Force (N) = $$\Delta $$P $$ \times $$ A

       = B $$ \times $$ $${{2\Delta a} \over a}$$ $$ \times $$ (2$$\pi $$a)b

       =   4$$\pi $$B$$\Delta $$ab

$$ \therefore $$   Frictional force = $$\mu $$N = (4$$\pi $$$$\mu $$Bb)$$\Delta $$a
2
MCQ (Single Correct Answer)

JEE Main 2017 (Offline)

A copper ball of mass 100 gm is at a temperature T. It is dropped in a copper calorimeter of mass 100 gm, filled with 170 gm of water at room temperature. Subsequently, the temperature of the system is found to be 75oC. T is given by: (Given : room temperature = 30oC, specific heat of copper = 0.1 cal/gmoC)
A
825oC
B
800oC
C
885oC
D
1250oC

Explanation

According to principle of calorimetry,

Heat lost = Heat gain

100 × 0.1( – 75) = 100 × 0.1 × 45 + 170 × 1 × 45

10 – 750 = 450 + 7650

10 = 1200 + 7650 = 8850

T = 885°C
3
MCQ (Single Correct Answer)

JEE Main 2017 (Offline)

A man grows into a giant such that his linear dimensions increase by a factor of 9. Assuming that his density remains same, the stress in the leg will change by a factor of:
A
$${1 \over {81}}$$
B
9
C
$${1 \over {9}}$$
D
81

Explanation

Stress = $${{Force} \over {Area}}$$ = $${{mg} \over A}$$ = $${{\rho Vg} \over A}$$ $$\left(As\ {\rho = {m \over V}} \right)$$

$$ \Rightarrow $$ Stress $$ \propto $$ $${{{L^3}} \over {{L^2}}}$$

$$ \Rightarrow $$ Stress $$ \propto $$ L

As linear dimension increases by a factor of 9, so stress also increases by a factor of 9.
4
MCQ (Single Correct Answer)

JEE Main 2017 (Offline)

An external pressure P is applied on a cube at 0oC so that it is equally compressed from all sides. K is the bulk modulus of the material of the cube and $$\alpha$$ is its coefficient of linear expansion. Suppose we want to bring the cube to its original size by heating. The temperature should be raised by:
A
$${P \over {3\alpha K}}$$
B
$${P \over {\alpha K}}$$
C
$${3 \alpha \over {P K}}$$
D
3PK$$\alpha$$

Explanation

As we know, Bulk modulus

K = $${{\Delta P} \over {\left( {{{ - \Delta V} \over V}} \right)}}$$

$$ \Rightarrow $$ $${{{\Delta V} \over V} = {P \over K}}$$

V = V0(1 + $$\gamma $$$$\Delta $$t)

$${{{\Delta V} \over {{V_0}}} = \gamma \Delta t}$$

$$ \therefore $$ $${{P \over K} = \gamma \Delta t}$$

$$ \Rightarrow $$ $${\Delta t = {P \over {\gamma K}}}$$ = $${{P \over {3\alpha K}}}$$

Questions Asked from Properties of Matter

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