TS Inter 1st Year Physics Study Material Chapter 6 Work, Energy and Power

Telangana TSBIE TS Inter 1st Year Physics Study Material 6th Lesson Work, Energy and Power Textbook Questions and Answers.

TS Inter 1st Year Physics Study Material 6th Lesson Work, Energy and Power

Very Short Answer Type Questions

Question 1.
If a bomb at rest explodes into two pieces, the pieces must travel in opposite directions. Explain.
Answer:
Explosion is due to internal forces. In law of conservation of linear momentum internal forces cannot change the momentum of the system. So after explosion m1v1 + m2v2 = 0 or m1v1 = – m2v2 ⇒ they will fly in opposite directions.

Question 2.
State the conditions under which a force does no work.
Answer:

  1. When force (F) and displacement (S) are mutually perpendicular then work done is zero.
    ∵ W = \(\overline{\mathrm{F}}.\overline{\mathrm{S}}\) = |F| |S| cos θ when θ = 90° work W = 0
  2. Even though force is applied if displacement is zero then work done W = 0.

Question 3.
Define Work, Power and Energy. State their S.I. units.
Answer:
Work :
The product of force and displacement along the direction of force is called work.
Work done W = \(\overline{\mathrm{F}}.\overline{\mathrm{S}}\)
= \(|\overline{\mathrm{F}}||\overline{\mathrm{S}}|\) cos θ
S.I. unit of work is Joule.
Dimensional formula : ML²T-2.

Power :
The rate of doing work is called power.
TS Inter 1st Year Physics Study Material Chapter 6 Work, Energy and Power 1
S.I. unit: Watt;
D.F. : ML²T-3

Energy :
It is the capacity or ability of the body to do work. By spending energy we can do work or by doing work energy contentment of the body will increase.
S.I. unit: Joule ; D.F. : MML²T-2

Question 4.
State the relation between the kinetic energy and momentum of a body.
Answer:
Kinetic energy K.E = \(\frac{1}{2}\)mv² ;
momentum \(\overline{\mathrm{p}}\) = mv
Relation between P and KE is
K.E = p²/2m. ⇒ P = \(\sqrt{K.E.2m}\)

TS Inter 1st Year Physics Study Material Chapter 6 Work, Energy and Power

Question 5.
State the sign of work done by a force in the following.
a) Work done by a man in lifting a bucket out of a well by means of a rope tied to the bucket.
b) Work done by gravitational force in the above case.
Answer:
a) When a bucket is lifted out of well work is done against gravity so work done is negative.

b) Work done by gravitational force is positive.

Question 6.
State the sign of work done by a force in the following.
a) work done by friction on a body sliding down an inclined plane.
b) work done gravitational force in the above case.
Answer:
a) Work done by friction while sliding down is negative. Because it opposes downward motion of the body.

b) Work done by gravitational force when a body is sliding down is positive.

Question 7.
State the sign of work done by a force in the following.
a) work done by an applied force on a body moving on a rough horizontal plane with uniform velocity.
b) work done by the resistive force of air on a vibrating pendulum in bringing it to rest.
Answer:
a) Work done against the direction of motion of a body moving on a horizontal plane is negative.

b) In pendulum a∝ – y. So work done by air resistance to bring it to rest is considered as positive.

Question 8.
State if each of the following statements is true or false. Give reasons for your answer.
a) Total energy of a system is always conserved, no matter what internal and external forces on the body are present
b) The work done by earth’s gravitational force in keeping the moon in its orbit for its one revolution is zero.
Answer:
a) Law of conservation of energy states that energy can be neither created nor destroyed. This rule is applicable to internal forces and also for external forces when they are conservative forces.

b) Gravitational forces are conservative forces. Work done by conservative force around a closed path is zero.

Question 9.
Which physical quantity remains constant (i) in an elastic collision (ii) in an inelastic collision?
Answer:
In elastic collision :
P and K.E. are conserved, (remains constant)

In inelastic collision :
only momentum is conserved, (remains constant)

Question 10.
A body freely falling from a certain height ‘h’, after striking a smooth floor rebounds and h rises to a height h/2. What is the coefficient of restitution between the floor and the body?
Answer:
Given that, h1 = h and h2 = \(\frac{h}{2}\)
We know that coefficient of restitution.
TS Inter 1st Year Physics Study Material Chapter 6 Work, Energy and Power 2

Question 11.
What is the total displacement of freely falling body, after successive rebounds from the same place of ground, before it comes to stop? Assume that V is the coefficient of restitution between the body and the ground.
Answer:
Total displacement of a freely falling body after successive rebounds from the same place of ground, before it comes to stop is equal to height (h) from which the body is dropped.

Short Answer Questions

Question 1.
What is potential energy? Derive an expression for the gravitational potential energy.
Answer:
Potential energy :
It is the energy possessed by a body by the virtue of its position.
Ex: Energy stored in water a over head tank, wound spring.
TS Inter 1st Year Physics Study Material Chapter 6 Work, Energy and Power 3

Equation for potential energy :
Let a body of mass m is lifted through a height ‘h’ above the ground. Where ground is taken as refe-rence. In this process we are doing some work.

Work done against gravity W = m.g.h.
i. e., Force × displacement along the direction of force applied. This work done is stored in the body in the form of potential energy. Because work and energy can be interchanged.
∴ Potential Energy P.E. = mgh.

TS Inter 1st Year Physics Study Material Chapter 6 Work, Energy and Power

Question 2.
A lorry and a car moving with the same momentum are brought to rest by the application of brakes, which provide equal retarding forces. Which of them will come to rest in shorter time? Which will come to rest in less distance?
Answer:
Momentum (\(\overline{\mathrm{P}}\) = mv) is same for both lorry and car.
Work done to stop a body = Kinetic energy stored
∴ W = F. S = \(\frac{1}{2}\) mv² = K.E. But force applied by brakes is same for lorry and car.
Relation between \(\overline{\mathrm{P}}\) on K.E. is
TS Inter 1st Year Physics Study Material Chapter 6 Work, Energy and Power 4

So lighter body (car) will travel longer distance when P, F are same.
Then mS = constant.
∴ So car travels longer distance than lorry before it is stopped.

Question 3.
Distinguish between conservative and non-conservative forces with one example each.
Answer:
i) Conservative forces :
If work done by the force around a closed path is zero and it is independent of the path then such forces are called conservative forces.

Example:

  1. Work done in lifting a body in gravitational field. When the body returns to its original position work done on it is zero.
    So gravitational forces are conservative forces.
  2. Let a charge ‘q’ is moved in an electric field on a closed path then change in its electric potential i.e., static forces are conservative forces.

ii) Non-conservative forces :
For non-conservative forces work done by a force around a closed path is not equal to zero and it is dependent on the path.
Ex: Work done to move a body against friction. While taking a body between two points say A & B. We have to do work to move the body from A to B and also work is done to move the body from B to A. As result, the work done in moving the body in a closed path is not equals to zero. So frictional forces are non-conservative forces.

Question 4.
Show that in the case of one dimensional elastic collision, the relative velocity of approach of two colliding bodies before collision is equal to the relative velocity of separation after collision.
Answer:
To show relative velocity of approach of two colliding bodies before collision is equal to relative velocity of separation after collision.

Let two bodies of masses m1, m2 are moving with velocities u1, u2 along the straight line in same direction collided elastically.

Let their velocities after collision be v1 and v2.
TS Inter 1st Year Physics Study Material Chapter 6 Work, Energy and Power 5

According to the law of conservation of linear momentum
m1u1 + m2u2 = m1v1 + m2v2 or m1(u1 – v1) = m2(v2 – u2) ………… (1)

According to law of conservation of kinetic energy
\(\frac{1}{2}\)m11 + \(\frac{1}{2}\)m2u²2 = \(\frac{1}{2}\)m12 + \(\frac{1}{2}\)m22
m1(u²1 – v²1) = m2(v²2 – u²2) ………. (2)
Dividing eqn. (2) by (1)
TS Inter 1st Year Physics Study Material Chapter 6 Work, Energy and Power 6
u1 + v1 = v2 + u2 ⇒ u1 – u2 = v2 – v1 ……. (3)
i.e., relative velocity of approach of the two bodies before collision = relative velocity of separation of the two bodies after collision. So coefficient of restitution is equal to ‘1’.

Question 5.
Show that two equal masses undergo oblique elastic collision will move at right angles after collision, if the second body initially at rest.
Answer:
Consider two bodies possess equal mass (m) and they undergo oblique elastic collision.

Let the first body moving with initial velocity ‘u’ collides with the second body at rest.
TS Inter 1st Year Physics Study Material Chapter 6 Work, Energy and Power 7

In elastic collision, momentum is conserved. So, conservation of momentum along X-axis yields.
mu = mv1 cos θ1 + mv2 cos θ2.
(i.e.) u = v1 cos θ1 + v2 cos θ2 ……. (1)
along Y-axis
0 = v1 sin θ1 – v2 sin θ2 ……… (2)
squaring and adding eq. (1) and (2) we get
u² = v²1 + v²2 + 2v1v2 cos (θ1 + θ2) …. (3)
As the collision is elastic,
Kinetic Energy (K.E.) is also conserved.

From eq. (3) and (4) 2v1v2 cos(θ1 + θ2) = 0
As it is given that v1 ≠ 0 and v2 ≠ 0
∴ cos(θ1 + θ2) = 0 or θ1 + θ2 = 90°.
The two equal masses undergoing oblique elastic collision will move at right angles after collision, if the second body initially at rest.

TS Inter 1st Year Physics Study Material Chapter 6 Work, Energy and Power

Question 6.
Derive an expression for the height attained by a freely falling body after ‘n’ number of rebounds from the floor.
Answer:
Let a small ball be dropped from a height ‘h’ on a horizontal smooth plate. Let it rebounds to a height ‘h1‘.
TS Inter 1st Year Physics Study Material Chapter 6 Work, Energy and Power 8

Velocity with which it strikes the plate u1 = \(\sqrt{2gh}\)
Velocity with which it leaves the plate v1 = \(\sqrt{2gh_1}\)

The velocity of plate before and after collision is zero i.e., u2 = 0, v2 = 0
Coefficient of restitution,
TS Inter 1st Year Physics Study Material Chapter 6 Work, Energy and Power 9
For 2nd rebound it goes to a height
h2 = e²h1 = e²e²h = e4h

For 3rd rebound it goes to a height
h3 = e²h2 = e²e4h = e6h

For nth rebound height attained
hn = e2nh.

Question 7.
Explain the law of conservation of energy.
Answer:
Law of conservation of energy:
Wien forces doing work on a system are conservative then total energy of the system is constant i.e., energy can neither be created nor destroyed.
i.e., Total energy = (K + u) = constant form.

Explanation :
Consider a body undergoes small displacement ∆x under the action of conservative force F. According to work energy theorem.
Change in K.E = work done
∆K = F(x)∆x ………….. (1)
but Potential energy Au = -F(x)∆x ………….. (2)
from (1) and (2) = ∆K = – ∆u
⇒ ∆(K + u) = 0
Hence (K + u) = constant
i.e., sum of the kinetic energy and potential energy of the body is a constant

Since the universe may be considered as an isolated system, the total energy of the universe is constant.

Long Answer Questions

Question 1.
Develop the notions of work and kinetic energy and show that it leads to work- energy theorem. State the conditions under which a force does no work. [AP Mar. I 7, 15, May 1 7; TS Mar. 15]
Answer:
Work :
The product of component of force in the direction of displacement and the magnitude of displacement is called work.
W = \(\overline{\mathrm{F}}.\overline{\mathrm{S}}\)
When \(\overline{\mathrm{F}}\) and \(\overline{\mathrm{S}}\) are parallel W = \(|\overline{\mathrm{F}}|\times|\overline{\mathrm{S}}|\)
When \(\overline{\mathrm{F}}\) and \(\overline{\mathrm{S}}\) has some angle 6 between them
W = \(\overline{\mathrm{F}}.\overline{\mathrm{S}}\) cos θ
TS Inter 1st Year Physics Study Material Chapter 6 Work, Energy and Power 10

Kinetic energy :
Energy possessed by a moving body is called kinetic energy (k)

The kinetic energy of an object is a measure of the work that an object can do by the virtue of its motion.

Kinetic energy can be measured with equation K = \(\frac{1}{2}\)mv²
Ex : All moving bodies contain kinetic energy.

Work energy theorem (For variable force):
Work done by a variable force is always equal to the change in kinetic energy of the body.
Work done W = \(\frac{1}{2}\)mV² – \(\frac{1}{2}\)mV²0? = Kf – Ki

Proof :
Kinetic energy of a body K = \(\frac{1}{2}\)mv²
Time rate of change of kinetic energy is
TS Inter 1st Year Physics Study Material Chapter 6 Work, Energy and Power 11

When force is conservative force F = F(x)
∴ On integration over initial position (x1) and final position x2
TS Inter 1st Year Physics Study Material Chapter 6 Work, Energy and Power 12

i.e., work done by a conservative force is equal to change in kinetic energy of the body.
Condition for Force not to do any work.

When Force (\(\overline{\mathrm{F}}\)) and displacement (\(\overline{\mathrm{S}}\)) are perpendicular work done is zero, i.e., when
θ = 90° then W = \(\overline{\mathrm{F}}.\overline{\mathrm{S}}\) = 0

TS Inter 1st Year Physics Study Material Chapter 6 Work, Energy and Power

Question 2.
What are collisions? Explain the possible types of collisions? Develop the theory of one dimensional elastic collision. [TS Mar.’ 18; AF Mar. 19. May 14]
Answer:
A process in which the motion of a system of particles changes but keeping the total momentum conserved is called collision.

Collisions are two types :

  1. elastic
  2. inelastic.

To show relative velocity of approach before collision is equal to relative velocity of separation after collision.

Let two bodies of masses m1, m2 are moving with velocities u1, u2 along the same line in same direction collided elastically.

Let their velocities after collision are v1 and v2.
TS Inter 1st Year Physics Study Material Chapter 6 Work, Energy and Power 13

According to the law of conservation of linear momentum
m1u1 + m2u2 = m1v1 + m2v2
or m1 ( u1 – v1 ) = m2 ( v2 – u2 ) ……… (1)

According to law of conservation of kinetic energy
TS Inter 1st Year Physics Study Material Chapter 6 Work, Energy and Power 14

i.e., In elastic collisions relative velocity of approach of the two bodies before collision = relative velocity of separation of the two bodies after collision.

Velocities of two bodies after elastic collision:
TS Inter 1st Year Physics Study Material Chapter 6 Work, Energy and Power 15

Question 3.
State and prove law of conservation of energy in case of a freely falling body. [TS Mar. ’19, ’17, ’16, May ’18, ’17, ’16, June ’15; AP Mar. ’18, ’16, ’15, May ’18, ’16, June ’15, May ’13]
Answer:
Law of conservation of energy :
Energy can neither be created nor destroyed. But it can be converted from one form into the another form so that the total energy will remains constant in a closed system.

Proof : In case of a freely fidling body :
Let a body of mass is dropped from a height H’ at point A.

Forces due to gravitational field are conservative forces, so total mechanical energy (E = P.E + K.E.) is constant i.e., neither destroyed nor created.
TS Inter 1st Year Physics Study Material Chapter 6 Work, Energy and Power 16

The conversion of potential energy to kinetic energy for a ball of mass ra dropped from a height H

1. At point H : Velocity of body v = 0
⇒ K = 0
Potential energy (u) = mgH
where H=height above the ground
T.E = u + K = mgH (1)

2. At point 0 :
i.e., just before touching the ground :
A constant force is a special case of specially dependent force F(x) so mechanical energy is conserved.
So energy at H = Energy at 0 = mgH

Proof:
At point ‘0’ height h = 0 ⇒
⇒ v = \(\sqrt{2gH}\) ; u = 0
K0 = \(\frac{1}{2}\) mv² = \(\frac{1}{2}\) m2gH = mgH
Total energy E = mgH + 0 = mgH ………….. (2)

3. At any point h:
Let height above ground = h
u = mgh, Kh = \(\frac{1}{2}\)mV²
where v = \(\sqrt{2g(h – x)}\)
∴ Velocity of the body when it falls through a height (h – x) is \(\sqrt{2g(h – x)}\)
∴ Total energy =mgh + \(\frac{1}{2}\)m2g(H – h)
⇒ E = mgh + mgH – mgh = mgH ………… (3)
From eq. 1, 2 & 3 total energy at any point is constant.
Hence, law of conservation of energy is proved.

Conditions to apply law of conservation of energy:

  1. Work done by internal forces is conservative.
  2. No work is done by external force.

When the above two conditions are satisfied then total mechanical energy of a system will remain constant.

Problems

Question 1.
A test tube of mass 10 grams closed with a cork of mass 1 gram contains some ether. When the test tube is heated the cork flies out under the presssure of the ether gas. The test tube is suspended horizontally by a weight less rigid bar of length 5 cm. What is the minimum velocity with which the cork should fly out of the tube, so that test tube describing a full vertical circle about the point O. Neglect the mass of ether.
Solution:
Length of bar, L = 5 cm, = \(\frac{5}{100}\), g = 10m/s²
For the cork not to come out minimum velocity at lowest point is, v = \(\sqrt{5gL}\). At this condition centrifugal and centripetal forces are balanced.
TS Inter 1st Year Physics Study Material Chapter 6 Work, Energy and Power 17

Question 2.
A machine gun fires 360 bullets per minute and each bullet travels with a velocity of 600 ms-1. If the mass of each bullet is 5 gm, find the power of the machine gun? [AP May ’16, ’13, June ’15, Mar. ’14; AP Mar. ’18. ’16; TS May ’18]
Solution:
Number of bullets, n = 360
Time, t = 1 minute = 60s
Velocty of the bullet, v = 600 ms-1 ; Mass of each bullet, m = 5gm = 5 × 10-3 kg
TS Inter 1st Year Physics Study Material Chapter 6 Work, Energy and Power 18
⇒ P = 5400W = 5.4KW

TS Inter 1st Year Physics Study Material Chapter 6 Work, Energy and Power

Question 3.
Find the useful power used in pumping 3425 m³ of water per hour from a well 8 m deep to the surface, supposing 40% of the horse power during pumping is wasted. What is the horse power of the engine?
Solution:
Mass of water pumped, m = 3425 m³
= 3425 × 10³ kg.
Mass of lm³ water = 1000 kg
Depth of well d = 8 m., Power wasted = 40%
∴ efficiency, η = 60%
time, t = 1 hour = 3600 sec.
TS Inter 1st Year Physics Study Material Chapter 6 Work, Energy and Power 19

Question 4.
A pump is required to lift 600 kg of water per minute from a well 25m deep and to eject it with a speed of 50 ms-1. Calculate the power required to perform the above task? (g = 10 m sec-2) [TS Mar. ’19, ’16; AP May 18, Mar. 15, June 15]
Solution:
Mass of water m = 600 kg; depth = h = 25 m
Speed of water v = 25 m/s; g = 10 m/s², time t = 1 min = 60 sec.
Power of motor P = Power to lift water (P1) + Kinetic energy of water (K.E) per second.
Power to lift water
TS Inter 1st Year Physics Study Material Chapter 6 Work, Energy and Power 20
∴ Power of motor P = 2500 + 3125 = 5625 watt 5.625 K.W.

Question 5.
A block of mass 5 kg initially at rest at the origin is acted on by a force along the X-positive direction represented by F=(20 + 5x)N. Calculate the work done by the force during the displacement of the block from x = 0 to x = 4m.
Solution:
Mass of block, m = 5 kg
Force acting on the block, F = (20 + 5x) N
If ‘w’ is the total amount of work done to displace the block from x = 0 to x = 4m then,
TS Inter 1st Year Physics Study Material Chapter 6 Work, Energy and Power 21

Question 6.
A block of mass 5 kg is sliding down a smooth inclined plane as shown. The spring arranged near the bottom of the inclined plane has a force constant 600 N/m. Find the compression in the spring at the moment the velocity of the block is maximum?
TS Inter 1st Year Physics Study Material Chapter 6 Work, Energy and Power 22
Solution:
Mass of the block, m = 5kg
Force constant, K = 600 N m-1
From figure, sin θ = \(\frac{3}{5}\)
Force produced by the motion in the block,
F = mg sin0 ⇒ F = 5 × 9.8 × \(\frac{3}{5}\) = 29.4 N
But force constant K = \(\frac{F}{x}\) x
∴ x = \(\frac{F}{K}=\frac{29.5}{600}\) = 0.05m = 5cm

Question 7.
A force F = – \(\frac{K}{x^2}\) (x ≠ 0) acts on a particle along the X-axis. Find the work done by the force in displacing the particle from x = + a to x = + 2a. Take K as a positive constant.
Solution:
Force acting on the particle, F = –\(\frac{K}{x^2}\)
Total amount of work done to displace the particle from x = + a to x = + 2a is,
TS Inter 1st Year Physics Study Material Chapter 6 Work, Energy and Power 23

TS Inter 1st Year Physics Study Material Chapter 6 Work, Energy and Power

Question 8.
A force F acting on a particle varies with the position x as shown in the graph. Find the work done by the force in displacing the particle from x = – a to x = + 2a?
TS Inter 1st Year Physics Study Material Chapter 6 Work, Energy and Power 24
Solution:
Average force acting on the particle, F = \(\frac{F}{K}\)

Amount of work done by the force to displace the particle from x = -a to x = +2a is,
TS Inter 1st Year Physics Study Material Chapter 6 Work, Energy and Power 25

Question 9.
From a height of 20 m above a horizontal floor, a ball is thrown down with initial velocity 20 m/s. After striking the floor, the ball bounces to the same height from which it was thrown. Find the coefficient of restitution for the collision between the ball and the floor? (g = 10 m/s²)
Solution:
Initial velocity = u¹ = 20 m/s, h – 20 m,
g = 10 m/s²

Velocity of approach,
u² = u = u +2as = 400+ 2 × 10 × 20
⇒ u² = 400 + 400 = 800 ⇒ u = 20√2
Height of rebounce = h = 20 m.
∴ Velocity of separation
TS Inter 1st Year Physics Study Material Chapter 6 Work, Energy and Power 26

Question 10.
A ball falls from a height of 10 m on to a hard horizontal floor and repeatedly bounces. If the coefficient of restitution is \(\frac{1}{\sqrt{2}\), then what is the total distance travelled by the ball before it ceases to rebound?
Solution:
Height from which the ball is allowed to fall, h = 10 m
Coefficient of restitution between the hard horizontal floor and the ball, e = \(\frac{1}{\sqrt{2}\)
∴ Total distance travelled by the ball before it ceases to rebound,
TS Inter 1st Year Physics Study Material Chapter 6 Work, Energy and Power 27

Question 11.
In a ballistics demonstration, a police officer fires a bullet of mass 50g with speed 200 msr1 on soft plywood of thickness 2 cm. The bullet emerges with only 10% of its initial kinetic energy. What is the emergent speed of the bullet?
Answer:
Mass of bullet m = 50g = 0.05 kg
Initial velocity V0 = 200 m/s
TS Inter 1st Year Physics Study Material Chapter 6 Work, Energy and Power 28

TS Inter 1st Year Physics Study Material Chapter 6 Work, Energy and Power

Question 12.
Find the total energy of a body of 5 kg mass, which is at a height of 10 in from the earth and foiling downwards straightly with a velocity of 20 m/s. (Take the acceleration due to gravity as 10 m/s²) [TS May ’16]
Answer:
Mass m = 5 kg; Height h = 10 m ; g = 10 m/s²
Velocity v = 20 m/s.
TS Inter 1st Year Physics Study Material Chapter 6 Work, Energy and Power 29

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