Work Energy and Power Test

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Work Energy and Power Test - 25

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Weekly Quiz Competition

Question 1
1 / -0

A stone of mass $$1\ kg$$ is tied of the end of a string $$1\ m$$ long. It is whirled in a vertical circle. If the velocity of stone at the top is $$4\ m/s$$. What is the tension in the string at the lowest point?
Take $$g = 10\ m/s^{2}$$

A
$$6\ N$$

B
$$66\ N$$

C
$$5.2\ N$$

D
$$76\ N$$

Solution

Let the velocity at lowest point is $$v$$ and at highest point be $$u$$, then according to law of conservation of energy.
energy at lowest point= energy at highest point.

$$\Rightarrow\dfrac{1}{2}mv^2=mg(2r)+\dfrac{1}{2}mu^2 \\ \Rightarrow

v=\sqrt{u^2+4rg}=\sqrt{4^2+(4\times 1 \times 10)}=\sqrt{56}$$
At lowest point,

$$\dfrac { m{ v }^{ 2 } }{ r } =T-mg$$

$$\Rightarrow T=mg+ \dfrac { m{ v }^{ 2 } }{ r }=mg+ \dfrac { m (\sqrt{56} )^{ 2 } }{ r }$$ $$\quad$$ $$(r=1)$$

$$\Rightarrow T=mg+56\ m=66 \times m =66 \times 1 = 66\ N $$.
Question 2
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A massive ball moving with a speed $$v$$ collide with a tiny ball having a very small mass , immediately after the impact the second ball will move at speed approximately equal to :

A
$$\infty$$

B
$$\dfrac{v}{2}$$

C
$$v$$

D
$$2v$$

Solution

In an elastic collision where the projectile is much more massive than the target, the velocity of the target particle after the collision will be about twice that of the projectile and the projectile velocity will be essentially unchanged. Hence, the second ball will move at a speed approximately equal to $$2v$$.
Question 3
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A spring of spring constant $$5\times 10^3N/m$$ is stretched initially by 5 cm from the unstretched position. Then the work required to stretch it further by another 5 cm is

A
$$18.75 J$$

B
$$25.00 J$$

C
$$6.25 J$$

D
$$12.50 J$$

Solution

$$\displaystyle W_1=\frac{1}{2 }\times5\times 10^3(0.05)^2$$
$$\displaystyle \Rightarrow W_2=\frac{1}{2 }\times5\times 10^3(0.10)^2$$
$$\displaystyle \therefore \Delta W= W_2 -W_1 =\frac{1}{2 }\times5\times 10^3(0.10)^2 -\frac{1}{2 }\times5\times 10^3(0.05)^2 \\ =\frac{1}{2}\times5\times 10^3\times 0.15\times 0.05=18.75J$$
Question 4
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Name the type of energy (kinetic energy $$K$$ or potential energy $$U$$) possessed in the following case.
A piece of stone placed on the roof.

A
$$U$$

B
$$K$$

C
$$U$$ and $$K$$

D
No energy

Solution

When a stone is placed at the roof it is at a certain height that is given by $$U = m\times g\times h$$

As we have a certain value for $$h$$ there would be some value for potential energy.

As the stone is at rest at the roof it will not have any kinetic energy as its velocity is zero, $$K = 1/2 (m\times v^{2})=0$$.

Hence, only potential energy will be there.
Question 5
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A particle moves under the effect of a force $$F=cx$$ from $$x=0$$ to $$x=x_1$$, the work done in the process is

A
$$cx_1^2$$

B
$$\displaystyle \frac{1}{2} cx_1^2$$

C
$$2cx_1^2$$

D
zero

Solution

$$\displaystyle W=\overset{x_1}{\underset{0}{\int}}Fdx=\overset{x_1}{\underset{0}{\int}}cxdx=\left [ \frac{1}{2}cx^2 \right ]_0^{x_1}$$
$$\displaystyle =\frac{1}{2}c(x_1^2-0)=\frac{1}{2}cx_1^2$$
Question 6
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A pump is used to lift $$500\ kg$$ of water from a depth of $$80\ m$$ in $$10\ s$$.
(Take $$g=10\ ms^{-2}$$). Calculate the work done by the pump.

A
$$16 \times 10^5J$$

B
$$4\times 10^5J$$

C
$$4\times 10^8J$$

D
$$2\times 10^5J$$

Solution

Given,
Mass of water lifted, $$m=500\ kg$$
Displacement, $$d=80\ m$$
Time taken, $$t=10\ s$$

Force, $$F = m\times g$$
$$F = 500\times 10$$
$$F= 5000\ N$$

Work done, $$W= F\times d $$
$$W = 5000\times 80$$
$$W = 4\times 10^{5}\ J$$.
Question 7
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Name the type of energy (kinetic energy $$K$$ or potential energy $$U$$) possessed in the following case.
The bob of a simple pendulum at its extreme position.

A
$$K$$

B
$$U$$

C
$$K$$ and $$U$$

D
No energy

Solution

Here simple pendulum is at a certain height and at extreme position it will not have any velocity. Hence , kinetic energy will be zero. Whereas $$ U = m\times g\times h$$.
As it is at certain height it has some value for $$h$$. Hence, it will have certain potential energy.
Question 8
1 / -0

Two solid rubber balls $$A$$ and $$B$$ having masses $$200\ \&\ 400\ \text{gm}$$ respectively are moving in opposite direction with velocity of $$A$$ equal to $$0.3\ \text{m/sec}.$$ After collision the two balls come to rest when the velocity of $$B$$ is :

A
$$0.15\ \text{m/sec}$$

B
$$1.5\ \text{m/sec}$$

C
$$-0.15\ \text{m/sec}$$

D
$$\text{None of these}$$

Solution

$$\text{Let the velocity of 2nd ball be}\ v\ \text{m/s}$$
$${ m }_{ A }=0.2\ \text{kg};\quad { v }_{ A }=0.3\ \text{m/s};\quad { m }_{ 2 }=0.4\ \text{kg};\quad { v }_{ 2 }=v$$
$$\text{Applying Momentum Conservation:}$$
$$\text{Initial Momentum}=\text{Final Momentum}$$
$$\Rightarrow { m }_{ A }{ v }_{ A }+{ m }_{ B }{ v }_{ B }=0\quad (\text{final momentum})\\ \Rightarrow 0.2\times 0.3+0.4\times v=0\\ \Rightarrow -0.06/0.4=v\\ \Rightarrow v=-0.15\ \text{m/s}$$
Question 9
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A man raises a box of mass $$50 \ kg$$ to a height of $$2 \ m$$ in $$2 \ minutes$$, while another man raises the same box to the same height in $$5 \ minutes$$. What is the ratio of work done by them ?

A
$$1 : 1$$

B
$$2 : 1$$

C
$$1 : 2$$

D
$$4 : 1$$

Solution

$$Work = Fs\cos\theta$$
where, $$F$$ is the force applied, $$s$$ is the displacement, and $$\theta$$ is the angle between the force applied and displacement

Hence, work done is independent of time taken.

In the given cases, $$\theta = 0^{\circ}$$ as the force applied are in the same direction. Also, $$F=mg$$
So $$W=Fs=mgs$$

In both the cases, mass and displacement are the same.
$$W =50\times 10\times 2 = 1000 J$$

Hence work done is in the ratio of $$1000 : 1000 = 1:1$$.
Question 10
1 / -0

A spring of force constant $$800\ N/m$$ has an extension of $$5\ cm$$. The work done in extending it from $$5\ cm$$ to $$15\ cm$$ is

A
$$16\ J$$

B
$$8\ J$$

C
$$32\ J$$

D
$$24\ J$$

Solution

$$\displaystyle Workdone, W=\frac{1}{2}k\left ( x_2^2-x_1^2 \right )$$

$$\displaystyle =\frac{1}{2}k\left [ (0.15)^2-(0.05)^2 \right ]$$

$$\displaystyle =\frac{1}{2}\times 800 \times 0.02=8J$$

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Re-Attempt Weekly Quiz Competition

Work Energy and Power Test - 25

Result

Work Energy and Power Test - 25

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SHARING IS CARING

Question 1

$$6\ N$$

$$66\ N$$

$$5.2\ N$$

$$76\ N$$

Solution

Question 2

$$\infty$$

$$\dfrac{v}{2}$$

$$v$$

$$2v$$

Solution

Question 3

$$18.75 J$$

$$25.00 J$$

$$6.25 J$$

$$12.50 J$$

Solution

Question 4

$$U$$

$$K$$

$$U$$ and $$K$$

No energy

Solution

Question 5

$$cx_1^2$$

$$\displaystyle \frac{1}{2} cx_1^2$$

$$2cx_1^2$$

zero

Solution

Question 6

$$16 \times 10^5J$$

$$4\times 10^5J$$

$$4\times 10^8J$$

$$2\times 10^5J$$

Solution

Question 7

$$K$$

$$U$$

$$K$$ and $$U$$

No energy

Solution

Question 8

$$0.15\ \text{m/sec}$$

$$1.5\ \text{m/sec}$$

$$-0.15\ \text{m/sec}$$

$$\text{None of these}$$

Solution

Question 9

$$1 : 1$$

$$2 : 1$$

$$1 : 2$$

$$4 : 1$$

Solution

Question 10

$$16\ J$$

$$8\ J$$

$$32\ J$$

$$24\ J$$

Solution

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