Question:
Mathematically show that law of conservation of energy holds true for a free falling object. Consider negligible air resistance.
Answer:
According to law of conservation of energy,
Total Mechanical Energy (E) = Kinetic energy (Ek)+ Potential Energy (Ep) = constant
Let a body of mass m falls from a point A, which is at a height h from the ground as shown.
At A:
velocity = 0
height = h
Kinetic Energy Ek = 0
Potential Energy Ep = mgh
Total Energy (E) = Ek + Ep = 0 + mgh = mgh ....(I)
At B:
Total Displacement (S) = x
Height from ground = h - x
According to law of motion, v2 - u2 = 2aS
⇒ v2 - 0 = 2gx (initial velocity = 0)
v2 = 2gx
Kinetic Energy Ek = ½ mv2 = ½ × m × 2gx
⇒ Ek = mgx
Potential Energy Ep = mg (h -x)
∴ Total Energy (E) = Ek + Ep = mgx + mg(h -x) = mgx + mgh - mgx = mgh ...(II)
At C:
Height w.r.t. ground = 0
Displacement from top (S) = h
According to law of motion, v2 - u2 = 2aS
⇒ v2 - 0 = 2gh
v2 = 2gh
Kinetic Energy Ek = ½ mv2 = ½ × m × 2gh
⇒ Ek = mgh
Potential Energy Ep = mg × 0 = 0
∴ Total Energy (E) = Ek + Ep = mgh + 0 = mgh ...(III)
Since equations I and II and III are same, we conclude under the force of gravity, the mechanical energy of a body remains constant.
Work and Energy
Q 1.
A body is thrown vertically upwards. Its velocity goes on decreasing. Write the change in kinetic energy when its velocity becomes zero.
Q 2.
A 60 kg person climbs stairs of total height 20 m in 2 min.Calculate the power delivered.
Q 3.
What do the machines need for their working ?
Q 4.
How does the wind move the blades of a wind-mill ?
Q 5.
Flowing water can rotate a turbine.What type of energy is used up by the turbine ?
Q 6.
Define potential energy.Derive an expression for the potential energy of a body of mass m,at a height h above the surface of the Earth.
Q 7.
Name a machine that transforms muscular energy into useful mechanical work.
Q 8.
What is the form of energy possessed by a running car?
Q 9.
An object of mass 15 kg is moving with a uniform velocity of 4 m/s.What is the kinetic energy possessed by the object ?
Q 10.
Two girls,each of weight 400 N,climb up a rope through a height of 8 m.We name one of the girls A and the other B.Girl A takes 20 s while B takes 50 s to accomplish this task.What is the power expended by each girl ?
Q 11.
A boy of mass 55 kg runs up a flight of 40 stairs,each measuring 0.15 m.Calculate the work done by the boy
Q 12.
A force of 7 N acts on an object.The displacement is,say 8 m,in the direction of the force,[in Fig.].Let us take it that the force acts on the object throughout the displacement What is the work done in this case ?
Q 13.
Name two forms of mechanical energy.
Q 14.
A freely falling object eventually stops on reaching the ground.What happens to its kinetic energy ?
Q 15.
Why do some engines require fuels like petrol and diesel ?
Q 16.
What is the work done by a constant force ?When is it positive,negative or zero ?
Q 17.
The velocity of a body moving in a straight line is increased by applying a constant force F,for some distance in the direction of the motion. Prove that the increase in the kinetic energy of the body is equal to the work done by the force on the body.
Q 18.
Is a person doing any work by holding a suitcase ?
Q 19.
A coolie carries a load of 50 kg on his head and walks on a level road upto 100 m.What is the work done by him ?
Q 20.
Can there be displacement of an object in the absence of any force acting on it ? Think.Discuss this question with your friends and teacher.
Q 21.
Does the transfer of energy take place when you push a huge rock with all your might and fail to move it? Where is the energy you spend going?
Q 22.
The mass of a ball A is double the mass of another ball B.The ball A moves at half the speed of the ball B.Calculate the ratio of the kinetic energy of A to the kinetic energy of B.
Q 23.
Fill in the blanks :
Ability to do work is called _______.
Q 24.
Name the device which converts electrical energy into mechanical energy.
Q 25.
Can a body have energy without having momentum ?
Q 26.
How much work is done on a body of mass 1kg whirling on a circular path of radius 5m?
Q 27.
Name the type of energy possessed by a raised hammer?
Q 28.
Write an expression for kinetic energy of an object
Q 29.
A body of mass 15 kg undergoes downward displacement of 40m under the effect of gravitational force. How much work is done? (take g = 10 m/s
2)
Q 30.
The work done by the heart is I J per beat.Calculate the power of the heart if it beats 72 times/min.
Q 31.
When is work done by a force zero ?
Q 32.
Is energy a vector quantity ?
Q 33.
What do you mean by power of an agent ? Give an expression for power.
Q 35.
Fill in the blanks :
A form of energy an object has because of its motion is called ________.
Q 37.
What is the kinetic energy of an object ?
Q 38.
What are the various energy transformations that occur when you are riding a bicycle ?
Q 39.
What do you mean by work ?Give its scientific conception.
Q 40.
Define kinetic energy.Give some examples of bodies possessing kinetic energy.
Q 41.
Fill in the blanks :
Work done against a force is stored as _____.
Q 42.
A pendulum of length 1m and with a bob of mass 1 kg swings in a vertical plane. Find the work done (from the below said choices) by weight of the bob when it moves from position A to B (as shown in figure). (Take g = 10 m/sec^2)
a. 4J
b. -4J
c. -5 J
d. 5 J
Q 43.
A spring is stretched, will the work done by the stretching force positive or negative or zero.
Q 44.
Is work a scalar or vector quantity?
Q 45.
A battery lights a bulb. Describe the energy changes involved in the process.
Q 46.
Write an expression for the kinetic energy of an object in terms (i) velocity (
v) of an object (ii) in terms of momentum (
p) of an object.
Q 47.
A mass of 10 kg is at a point A on a table. It is moved to a point B. If the line joining A and B is horizontal, what is the work done on the object by the gravitational force? Explain your answer.
Q 48.
What are other forms of energies apart from mechanical energy? Does Law of Conservation of Energy apply to all of them?
Q 49.
An object of mass 40 kg is raised to a height of 5 m above the ground. What is its potential energy? If the object is allowed to fall, find its kinetic energy when it is half-way down.
Q 50.
A certain household has consumed 250 units of energy during a month. How much energy is this in joules?
