Work Power & Energy : Live Session 5 August 2020

A spring $40~\text {mm}$ long is stretched by the application of a force. If $10 ~\text{N}$ force required to stretch the spring through $1 ~\text{mm,}$ then work done in stretching the spring through $40 ~\text{mm}$ is
1. $84~\text{J}$
2. $68~\text{J}$
3. $23~\text{J}$
4. $8~\text{J}$

A particle of mass 10 kg is moving with velocity of $10 \sqrt{x}$ m/s, where x is displacement . The work done by net force during the displacement of particle from x = 4 to x = 9 m is
1. 1250 J
2. 1000 J
3. 3500 J
4. 2500 J

A body is displaced from (0,0) to (1m,1m) along the path x=y by a force $F = (x^{2} \hat{j} + y \hat{i}) N$ . The work done by this force will be :
1. $\frac{4}{3} J$
2. $\frac{5}{6} J$
3. $\frac{3}{2} J$
4. $\frac{7}{5} J$

In the figure shown, the potential energy U of a particle is plotted against its position 'x' from the origin. Then which of the following statement is correct?

1. $x_{1}$ is in stable equilibrium
2. $x_{2}$ is in stable equilibrium
3. $x_{3}$ is in stable equilibrium
4. none of these

A weightless rod of length 2l carries two equal mass 'm', one tied at lower end A and the other at the middle of the rod at B. The rod can rotate in a vertical plane about a fixed horizontal axis passing through C. The rod is released from rest in the horizontal position. The speed of the mass B at the instant rod becomes vertical is:

1. $\sqrt{\frac{3 g l}{5}} $
2. $\sqrt{\frac{4 g l}{5}} $
3. $\sqrt{\frac{6 g l}{5}} $
4. $\sqrt{\frac{7 g l}{5}} $

Potential energy $(U)$ related to coordinates is given by; $U=3(x+y).$ Work done by the conservative force when the particle is going from $(0,0), (2,3)$ is:
1. $15$ J
2. $-15$ J
3. $12$ J
4. $10$ J

A body of mass m hangs at one end of a string of length l, the other end of which is fixed. It is given a horizontal velocity so that the string would just reach where it makes an angle of 60° with the vertical. The tension in the string at mean position is
1. 2 mg
2. mg
3. 3 mg
4. $\sqrt{3} m g$

A small block is shot into each of the four tracks as shown below. Each of the tracks rises to the same height. The speed with which the block enters the track is the same in all cases. At the highest point of the track, the normal reaction is maximum in:

1.
2.
3.
4.	Same in all cases

Work done by a frictional force is
1. Negative
2. Positive
3. Zero
4. All of the above

10.

A force acts on a 30 gm particle in such a way that the position of the particle as a function of time is given by $x = 3 t - 4 t^{2} + t^{3}$ , where x is in metres and t is in seconds. The work done during the first 4 seconds is
(1) 5.28 J
(2) 450 mJ
(3) 490 mJ
(4) 530 mJ

11.

A force $\vec{F} = 5 \hat{i} + 6 \hat{j} - 4 \hat{k}$ acting on a body, produces a displacement $\vec{s} = 6 \vec{i} + 5 \vec{k} .$ Work done by the force is
(1) 18 units
(2) 15 units
(3) 12 units
(4) 10 units

12.

A particle moves from position ${\vec{r}}_{1} = 3 \hat{i} + 2 \hat{j} - 6 \hat{k}$ to position ${\vec{r}}_{2} = 14 \hat{i} + 13 \hat{j} + 9 \hat{k}$ under the action of force $4 \hat{i} + \hat{j} + 3 \hat{k} N .$ The work done will be
(1) 100 J
(2) 50 J
(3) 200 J
(4) 75 J

13.

A force $(\vec{F}) = 3 \hat{i} + c \hat{j} + 2 \hat{k}$ acting on a particle causes a displacement: $(\vec{s}) = - 4 \hat{i} + 2 \hat{j} + 3 \hat{k}$ in its own direction. If the work done is 6 J then the value of ‘c’ is
(1) 0
(2) 1
(3) 6
(4) 12

14.

A cord is used to lower vertically a block of mass M by a distance d with constant downward acceleration $\frac{g}{4}$ . Work done by the cord on the block is
(1) $M g \frac{d}{4}$
(2) $3 M g \frac{d}{4}$
(3) $- 3 M g \frac{d}{4}$
(4) Mgd

15.

Two springs have their force constant as k₁ and $k_{2} (k_{1} > k_{2})$ . When they are stretched by the same force
(1) No work is done in case of both the springs
(2) Equal work is done in case of both the springs
(3) More work is done in case of second spring
(4) More work is done in case of first spring

16.

The potential energy of a certain spring when stretched through a distance ‘S’ is 10 joule. The amount of work (in joule) that must be done on this spring to stretch it through an additional distance ‘S’ will be:
(1) 30
(2) 40
(3) 10
(4) 20

17.

The potential energy between two atoms in a molecule is given by $U\left ( x \right )=\frac{a}{x^{12}}-\frac{b}{x^{6}};$ where $a$ and $b$ are positive constants and $x$ is the distance between the atoms. The atoms are in stable equilibrium when:
1. $x=\sqrt[6]{\frac{11a}{5b}}$
2. $x=\sqrt[6]{\frac{a}{2b}}$
3. $x=0$
4. $x=\sqrt[6]{\frac{2a}{b}}$

18.

Two identical cylindrical vessels with their bases at the same level each contain a liquid of density; $ρ$ . The height of the liquid in one vessel is h₁ and that in the other vessel is h₂ $(h_{1} > h_{2})$ .The area of either base is A. The work done by gravity in equalizing the levels when the two vessels are connected is :
(1) $(h_{1} - h_{2}) g ρ$
(2) $(h_{1} - h_{2}) g A ρ$
(3) $\frac{1}{2} {(h_{1} - h_{2})}^{2} g A ρ$
(4) $\frac{1}{4} {(h_{1} - h_{2})}^{2} g A ρ$

19.

A spherical ball of mass $20$ kg is stationary at the top of a hill of height$100$ m. It slides down a smooth surface to the ground, then climbs up another hill of height $30$ m and finally slides down to a horizontal base at a height of $20$ m above the ground. The velocity attained by the ball is:
1. $10 $ m/s
2. $10 \sqrt{30} $ m/s
3. $40 $ m/s
4. $20 $ m/s

20.

A uniform chain of length $L$ and mass $M$ is lying on a smooth table and one-third of its length is hanging vertically down over the edge of the table. If $g$ is the acceleration due to gravity, the work required to pull the hanging part on the table is:
1. $MgL$
2. $\dfrac{MgL}{3}$
3. $\dfrac{MgL}{9}$
4. $\dfrac{MgL}{18}$

21.

The displacement x of a particle moving in one dimension under the action of a constant force is related to the time t by the equation $t = \sqrt{x} + 3$ , where x is in meters and t is in seconds. The work done by the force in the first 6 seconds is
(1) 9 J
(2) 6 J
(3) 0 J
(4) 3 J

22.

A force $F = -k(y\hat i +x\hat j)$ (where $k$ is a positive constant) acts on a particle moving in the $xy\text-$plane. Starting from the origin, the particle is taken along the positive $x\text-$axis to the point $(a,0)$ and then parallel to the $y\text-$axis to the point $(a,a)$. The total work done by the force on the particle is:
1. $-2ka^2$
2. $2ka^2$
3. $-ka^2$
4. $ka^2$

23.

An open knife edge of mass 'm' is dropped from a height 'h' on a wooden floor. If the blade penetrates upto the depth 'd' into the wood, the average resistance offered by the wood to the knife edge is
(1) mg
(2) $m g (1 - \frac{h}{d})$
(3) $m g (1 + \frac{h}{d})$
(4) $m g {(1 + \frac{h}{d})}^{2}$

24.

The relationship between force and position is shown in the given figure (in a one-dimensional case). The work done by the force in displacing a body from $x = 1~\text{cm}$ to $x = 5~\text{cm}$ is:

1.   $20~\text{ergs}$
2.   $60~\text{ergs}$
3.   $70~\text{ergs}$
4.   $700~\text{ergs}$

25.

Adjacent figure shows the force-displacement graph of a moving body, the work done in displacing body from x = 0 to x = 35 m is equal to-

(1) 50 J
(2) 25 J
(3) 287.5 J
(4) 200 J

26.

The graph between the resistive force $F$ acting on a body and the distance covered by the body is shown in the figure. The mass of the body is $25$ kg and the initial velocity is $2$ m/s. When the distance covered by the body is $4$ m, its kinetic energy would be:

1. $50$ J
2. $40$ J
3. $20$ J
4. $10$ J

27.

A particle is given a constant horizontal velocity from height h. Taking g to be constant every where, kinetic energy E of the particle w. r. t. time t is correctly shown in
(1)
(2)
(3)
(4)

28.

A particle which is constrained to move along the x-axis, is subjected to a force in the same direction which varies with the distance x of the particle from the origin as $F (x) = - k x + a x^{3}$ . Here k and a are positive constants. For $x \geq 0$ , the functional form of the potential energy U(x) of the particle is-
(1)
(2)
(3)
(4)

29.

The potential energy of a system is represented in the first figure. the force acting on the system will be represented by:

1.		2.
3.		4.

30.

The graph between $\sqrt{E}$ and $\frac{1}{p}$ is (E = kinetic energy and p = momentum)
1.
2.
3.
4.

31.

The force acting on a body moving along x-axis varies with the position of the particle as shown in the fig.

The body is in stable equilibrium at
1. x = x₁
2. x = x₂
3. both x₁ and x₂
4. neither x₁ nor x₂

32.

The potential energy of a particle varies with distance x as shown in the graph. The force acting on the particle is zero at

1. C
2. B
3. B and C
4. A and D

33.

The force required to stretch a spring varies with the distance as shown in the figure. If the experiment is performed with the above spring of half length, the line OA will

1. Shift towards F-axis
2. Shift towards X-axis
3. Remain as it is
4. Become double in length

34.

A body moves from rest with a constant acceleration. Which one of the following graphs represents the variation of its kinetic energy K with the distance travelled x ?
1.
2.
3.
4.

35.

The relationship between the force F and the position x of a body is as shown in the figure. The work done in displacing the body from x = 1 m to x = 5 m will be:

1.	30 J	2.	15 J
3.	25 J	4.	20 J

36.

Two particles of masses m₁,m₂ move with initial velocities u₁ and u₂. On collision, one of the particles get excited to higher level, after absorbing energy $ε$ . If final velocities of particles be v₁ and v₂, then we must have
1. m₁²u₁+m₂²u₂- $ε$ =m₁²v₁+m₂²v₂
2. $\frac{1}{2}$ m₁u₁²+ $\frac{1}{2}$ m₂u₂= $\frac{1}{2}$ m₁v₁²+ $\frac{1}{2}$ m₂v₂²- $ε$
3. $\frac{1}{2}$ m₁u₁²+ $\frac{1}{2}$ m₂u₂²- $ε$ = $\frac{1}{2}$ m₁v₁²+ $\frac{1}{2}$ m₂v₂²
4. $\frac{1}{2}$ m₁²u₁²+ $\frac{1}{2}$ m₂²u₂²+ $ε$ = $\frac{1}{2}$ m₁²v₁²+ $\frac{1}{2}$ m₂²v₂²

37.

The potential energy U between two molecules as a function of the distance X between them has been shown in the figure. The two molecules are -

1. Attracted when x lies between A and B and are repelled when X lies between B and C
2. Attracted when x lies between B and C and are repelled when X lies between A and B
3. Attracted when they reach B
4. Repelled when they reach B

38.

The points of maximum and minimum attraction in the curve between potential energy (U) and distance (r) of a diatomic molecules are respectively -

(1) S and R
(2) T and S
(3) R and S
(4) S and T

39.

Consider a drop of rainwater having a mass of $1~\text{gm}$ falling from a height of $1~\text{km}.$ It hits the ground with a speed of $50~\text{m/s}.$ Take $g$ as constant with a value $10~\text{m/s}^2.$ The work done by the
(i) gravitational force and the (ii) resistive force of air is:

1.	$(\text{i})~1.25~\text{J};$ $(\text{ii})~-8.25~\text{J}$
2.	$(\text{i})~100~\text{J};$ $(\text{ii})~8.75~\text{J}$
3.	$(\text{i})~10~\text{J};$ $(\text{ii})~-8.75~\text{J}$
4.	$(\text{i})~-10~\text{J};$ $(\text{ii})~-8.75~\text{J}$

40.

A block of mass M is connected to a massless pulley and massless spring of stiffness k. The pulley is frictionless. The spring connecting the block and spring is massless. Initially the spring is untstretched when the block is released. When the spring is maximum stretched, then tension in the rope is

1. zero 2. Mg
3. 2Mg 4. Mg/2

41. Forces acting on a particle have magnitudes of $14,7,$ and $7$ N and act in the direction of vectors $6\hat{i} + 2\hat{j} + 3\hat{k}$, $3\hat{i} - 2\hat{j} + 6\hat{k}$, $2\hat{i} - 3\hat{j} - 6\hat{k}$ respectively. The forces remain constant while the particle is displaced from point $A$: $(2,1,-3)~\text{to}~B: (5,1,1)$. The coordinates are specified in meters. The work done equal to:

1.	$75~\text{J}$	2.	$55~\text{J}$
3.	$85~\text{J}$	4.	$65~\text{J}$

42.

A chain of length $L$ and mass $m$ is placed upon a smooth surface. The length of $(BA)$ is $(L–b).$ What will be the velocity of the chain when its end $A$ reaches $B?$

1. $ \sqrt{\dfrac{2 g \sin \theta}{L}\left(L^2-b^2\right)} $
2. $ \sqrt{\dfrac{g \sin \theta}{2 L}\left(L^2-b^2\right)} $
3. $ \sqrt{\dfrac{g \sin \theta}{L}\left(L^2-b^2\right)}$
4. None of these

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1.	\((\text{i})~1.25~\text{J};\) \((\text{ii})~-8.25~\text{J}\)
2.	\((\text{i})~100~\text{J};\) \((\text{ii})~8.75~\text{J}\)
3.	\((\text{i})~10~\text{J};\) \((\text{ii})~-8.75~\text{J}\)
4.	\((\text{i})~-10~\text{J};\) \((\text{ii})~-8.75~\text{J}\)

1.	\(75~\text{J}\)	2.	\(55~\text{J}\)
3.	\(85~\text{J}\)	4.	\(65~\text{J}\)

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