Joint Entrance Examination

Graduate Aptitude Test in Engineering

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1

MCQ (Single Correct Answer)

A point particle of mass $$m,$$ moves long the uniformly rough track $$PQR$$ as shown in the figure. The coefficient of friction, between the particle and the rough track equals $$\mu .$$ The particle is released, from rest from the point $$P$$ and it comes to rest at point $$R.$$ The energies, lost by the ball, over the parts, $$PQ$$ and $$QR$$, of the track, are equal to each other , and no energy is lost when particle changes direction from $$PQ$$ to $$QR$$.

The value of the coefficient of friction $$\mu $$ and the distance $$x$$ $$(=QR),$$ are, respectively close to:

A

$$0.29$$ and $$3.5$$ $$m$$

B

$$0.29$$ and $$6.5$$ $$m$$

C

$$0.2$$ and $$6.5$$ $$m$$

D

$$0.2$$ and $$3.5$$ $$m$$

Using work energy theorem for the motion of the particle,

Loss in $$P.E.=$$ Work done against friction from $$p \to Q$$

$$ + $$ work done against friction from $$Q \to R$$

$$mgh = \mu \left( {mg\cos \theta } \right)PQ + \mu mg\left( {QR} \right)$$

$$h = \mu \,\cos \,\theta \times PQ + \mu \left( {QR} \right)$$

$$2 = \mu \times {{\sqrt 3 } \over 2} \times {2 \over {\sin \,{{30}^ \circ }}} + \mu x$$

$$2 = 2\sqrt 3 \mu + \mu x$$ $$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,....(i)$$

[ $$\sin \,\,{30^ \circ } = {2 \over {PQ}}$$ ]

According to the question, work done $$P \to Q = $$ work done $$Q \to R$$

$$ \Rightarrow $$$$\mu \left( {mg\cos \theta } \right)PQ + \mu mg\left( {QR} \right)$$

$$\therefore$$ $$2\sqrt 3 \,\mu = \mu x$$

$$\therefore$$ $$x \approx 3.5m$$

From $$(i)$$

$$2 = 2\sqrt 3 \mu + 2\sqrt 3 \mu = 4\sqrt 3 \mu $$

$$\therefore$$ $$\mu = {2 \over {4\sqrt 3 }} = {1 \over {2 \times 1.732}} = 0.29$$

Loss in $$P.E.=$$ Work done against friction from $$p \to Q$$

$$ + $$ work done against friction from $$Q \to R$$

$$mgh = \mu \left( {mg\cos \theta } \right)PQ + \mu mg\left( {QR} \right)$$

$$h = \mu \,\cos \,\theta \times PQ + \mu \left( {QR} \right)$$

$$2 = \mu \times {{\sqrt 3 } \over 2} \times {2 \over {\sin \,{{30}^ \circ }}} + \mu x$$

$$2 = 2\sqrt 3 \mu + \mu x$$ $$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,....(i)$$

[ $$\sin \,\,{30^ \circ } = {2 \over {PQ}}$$ ]

According to the question, work done $$P \to Q = $$ work done $$Q \to R$$

$$ \Rightarrow $$$$\mu \left( {mg\cos \theta } \right)PQ + \mu mg\left( {QR} \right)$$

$$\therefore$$ $$2\sqrt 3 \,\mu = \mu x$$

$$\therefore$$ $$x \approx 3.5m$$

From $$(i)$$

$$2 = 2\sqrt 3 \mu + 2\sqrt 3 \mu = 4\sqrt 3 \mu $$

$$\therefore$$ $$\mu = {2 \over {4\sqrt 3 }} = {1 \over {2 \times 1.732}} = 0.29$$

2

MCQ (Single Correct Answer)

A person trying to lose weight by burning fat lifts a mass of $$10$$ $$kg$$ upto a height of $$1$$ $$m$$ $$1000$$ times. Assume that the potential energy lost each time he lowers the mass is dissipated. How much fat will he use up considering the work done only when the weight is lifted up? Fat supplies $$3.8 \times {10^7}J$$ of energy per $$kg$$ which is converted to mechanical energy with a $$20\% $$ efficiency rate. Take $$g = 9.8\,m{s^{ - 2}}$$ :

A

$$9.89 \times {10^{ - 3}}\,\,kg$$

B

$$12.89 \times {10^{ - 3}}\,kg$$

C

$$2.45 \times {10^{ - 3}}\,\,kg$$

D

$$6.45 \times {10^{ - 3}}\,\,kg$$

Assume the amount of fat is used = x kg

So total Mechanical energy available through fat

= $$x \times 3.8 \times {10^7} \times {{20} \over {100}}$$

And work done through lifting up

= 10 $$ \times $$ 9.8 $$ \times $$ 1000 = 98000 J

$$ \Rightarrow $$ $$x \times 3.8 \times {10^7} \times {{20} \over {100}}$$ = 98000

$$ \Rightarrow $$ $$x$$ = 12.89 $$ \times $$ 10^{-3} kg

So total Mechanical energy available through fat

= $$x \times 3.8 \times {10^7} \times {{20} \over {100}}$$

And work done through lifting up

= 10 $$ \times $$ 9.8 $$ \times $$ 1000 = 98000 J

$$ \Rightarrow $$ $$x \times 3.8 \times {10^7} \times {{20} \over {100}}$$ = 98000

$$ \Rightarrow $$ $$x$$ = 12.89 $$ \times $$ 10

3

MCQ (Single Correct Answer)

When a rubber-band is stretched by a distance $$x$$, it exerts restoring force of magnitude $$F = ax + b{x^2}$$ where $$a$$ and $$b$$ are constants. The work done in stretching the unstretched rubber-band by $$L$$ is :

A

$$a{L^2} + b{L^3}$$

B

$${1 \over 2}\left( {a{L^2} + b{L^3}} \right)$$

C

$${{a{L^2}} \over 2} + {{b{L^3}} \over 3}$$

D

$${1 \over 2}\left( {{{a{L^2}} \over 2} + {{b{L^3}} \over 3}} \right)$$

Given Restoring force, F = ax + bx^{2}

Work done in stretching the rubber-band by a distance $$dx$$ is

$$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,dW = F\,dx = \left( {ax + b{x^2}} \right)dx$$

Intergrating both sides,

$$W = \int\limits_0^L {axdx + \int\limits_0^L {b{x^2}dx}}$$

= $$\left[ {a{{{x^2}} \over 2} + b{{{x^3}} \over 3}} \right]_0^L$$

= $${{a{L^2}} \over 2} + {{b{L^3}} \over 3}$$

Work done in stretching the rubber-band by a distance $$dx$$ is

$$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,dW = F\,dx = \left( {ax + b{x^2}} \right)dx$$

Intergrating both sides,

$$W = \int\limits_0^L {axdx + \int\limits_0^L {b{x^2}dx}}$$

= $$\left[ {a{{{x^2}} \over 2} + b{{{x^3}} \over 3}} \right]_0^L$$

= $${{a{L^2}} \over 2} + {{b{L^3}} \over 3}$$

4

MCQ (Single Correct Answer)

This question has Statement $$1$$ and Statement $$2.$$ Of the four choices given after the Statements, choose the one that best describes the two Statements.

If two springs $${S_1}$$ and $${S_2}$$ of force constants $${k_1}$$ and $${k_2}$$, respectively, are stretched by the same force, it is found that more work is done on spring $${S_1}$$ than on spring $${S_2}$$.
**STATEMENT 1:** If stretched by the same amount work done on $${S_1}$$, Work done on $${S_1}$$ is more than $${S_2}$$
**STATEMENT 2:** $${k_1} < {k_2}$$

A

Statement 1 is false, Statement 2 is true

B

Statement 1 is true, Statement 2 is false

C

Statement 1 is true, Statement 2 is true, Statement 2 is the correct explanation for Statement 1

D

Statement 1 is true, Statement 2 is true, Statement 2 is not the correct explanation for Statement 1

We know force (F) = kx

$$W = {1 \over 2}k{x^2}$$

$$W =$$ $${{{{\left( {kx} \right)}^2}} \over {2k}}$$ $$\,\,\,$$

$$\therefore$$ $$W = {{{F^2}} \over {2k}}$$ [ as $$F=kx$$ ]

When force is same then,

$$W \propto {1 \over k}$$

Given that, $${W_1} > {W_2}$$

$$\therefore$$ $${k_1} < {k_2}$$

**Statement-2 is true.**

For the same extension, x_{1}
= x_{2}
= x

Work done on spring S_{1} is W_{1} = $${1 \over 2}{k_1}x_1^2 = {1 \over 2}{k_1}{x^2}$$

Work done on spring S_{2} is W_{2} = $${1 \over 2}{k_2}x_2^2 = {1 \over 2}{k_2}{x^2}$$

$$ \therefore $$ $${{{W_1}} \over {{W_2}}} = {{{k_1}} \over {{k_2}}}$$

As $${k_1} < {k_2}$$ then $${W_1} < {W_2}$$

**So, Statement-1 is false.**

$$W = {1 \over 2}k{x^2}$$

$$W =$$ $${{{{\left( {kx} \right)}^2}} \over {2k}}$$ $$\,\,\,$$

$$\therefore$$ $$W = {{{F^2}} \over {2k}}$$ [ as $$F=kx$$ ]

When force is same then,

$$W \propto {1 \over k}$$

Given that, $${W_1} > {W_2}$$

$$\therefore$$ $${k_1} < {k_2}$$

For the same extension, x

Work done on spring S

Work done on spring S

$$ \therefore $$ $${{{W_1}} \over {{W_2}}} = {{{k_1}} \over {{k_2}}}$$

As $${k_1} < {k_2}$$ then $${W_1} < {W_2}$$

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