Energy

ENERGY

Energy is defined as the capacity to do work. In other words, work done is the manifestation of energy. That is why work and energy have the same dimension (ML²T^-2)

The important aspect of energy is that for an isolated system, the sum of all forms of energy i.e., the total energy remains the same in any process irrespective of whatever internal changes may take place. This means that the energy disappearing in one form reappears in another form. This is known as the law of conservation of energy. In this chapter we shall take up only the mechanical energy for discussion.

In a broader sense, mechanical energy is classified into two types

i.               Kinetic energy

ii.               Potential energy

The energy possessed by a body due to its motion is called kinetic energy. The energy possessed by the body by virtue of its position is called potential energy.

The SI unit of energy is the same as that of work done i.e., N m (or) joule.

The dimension of energy is also the same as that of work done. It is given by [ML²T^-2]. The other units of energy and their SI equivalent values are given in Table 4.2.

Kinetic energy

Kinetic energy is the energy possessed by a body by virtue of its motion. All moving objects have kinetic energy. A body that is in motion has the ability to do work. For example a hammer kept at rest on a nail does not push the nail into the wood. Whereas the same hammer when it strikes the nail, draws the nail into the wood as shown in Figure 4.7. Kinetic energy is measured by the amount of work that the body can perform before it comes to rest. The amount of work done by a moving body depends both on the mass of the body and the magnitude of its velocity. A body which is not in motion does not have kinetic energy.

Work-Kinetic Energy Theorem

Work and energy are equivalents. This is true in the case of kinetic energy also. To prove this, let us consider a body of mass m at rest on a frictionless horizontal surface.

The work (W) done by the constant force (F) for a displacement (s) in the same direction is,

The constant force is given by the equation,

The third equation of motion (refer section 2.10.3) can be written as,

The expression for kinetic energy:

The term ½ mv² in the above equation is the kinetic energy of the body of mass (m) moving with velocity (v).

Kinetic energy of the body is always positive. From equations (4.10) and (4.11)

The expression on the right hand side (RHS) of equation (4.12) is the change in kinetic energy (∆KE) of the body.

This implies that the work done by the force on the body changes the kinetic energy of the body. This is called work-kinetic energy theorem.

The work-kinetic energy theorem implies the following.

i.               If the work done by the force on the body is positive then its kinetic energy increases.

ii.               If the work done by the force on the body is negative then its kinetic energy decreases.

iii.               If there is no work done by the force on the body then there is no change in its kinetic energy, which means that the body has moved at constant speed provided its mass remains constant.

Relation between Momentum and Kinetic Energy

Consider an object of mass m moving with a velocity . Then its linear momentum is  and its kinetic energy, KE =  ½ mv² .

Multiplying both the numerator and denominator of equation (4.13) by mass, m

where  is the magnitude of the momentum. The magnitude of the linear momentum can be obtained by

Note that if kinetic energy and mass are given, only the magnitude of the momentum can be calculated but not the direction of momentum. It is because the kinetic energy and mass are scalars.

Potential Energy

The potential energy of a body is associated with its position and configuration with respect to its surroundings. This is because the various forces acting on the body also depends on position and configuration.

“Potential energy of an object at a point P is defined as the amount of work done by an external force in moving the object at constant velocity from the point O (initial location) to the point P (final location). At initial point O potential energy can be taken as zero.

where the limit of integration ranges from initial location point O to final location point P.

We have various types of potential energies. Each type is associated with a particular force. For example,

(i) The energy possessed by the body due to gravitational force gives rise to gravitational potential energy.

(ii) The energy due to spring force and other similar forces give rise to elastic potential energy.

(iii) The energy due to electrostatic force on charges gives rise to electrostatic potential energy.

We will learn more about conservative forces in the section 4.2.7. Now, we continue to discuss more about gravitational potential energy and elastic potential energy.

Potential energy near the surface of the Earth

The gravitational potential energy (U) at some height h is equal to the amount of work required to take the object from ground to that height h with constant velocity.

Let us consider a body of mass m being moved from ground to the height h against the gravitational force as shown in Figure 4.8.

The gravitational force g acting on the body is, g = - mg jˆ (as the force is in y direction, unit vector ˆj is used). Here, negative sign implies that the force is acting  vertically downwards. In order to move the body without acceleration (or with constant velocity), an external applied force _a equal in magnitude but opposite to that of gravitational force _g has to be applied on the body i.e., a = - g  . This implies that a  = + mg j^ˆ. The positive sign implies that the applied force is in vertically upward direction. Hence, when the body is lifted up its velocity remains unchanged and thus its kinetic energy also remains constant.

The gravitational potential energy (U) at some height h is equal to the amount of work required to take the object from the ground to that height h.

Since the displacement and the applied force are in the same upward direction, the angle between them, θ=0^o. Hence, cos0⁰ =1 and |_a| = mg and |d| = dr .

Note that the potential energy stored in the object is defined through work done by the external force which is positive. Physically this implies that the agency which is applying the external force is transferring the energy to the object which is then stored as potential energy. If the object is allowed to fall from a height h then the stored potential energy is converted into kinetic energy.

Elastic Potential Energy

When a spring is elongated, it develops a restoring force. The potential energy possessed by a spring due to a deforming force which stretches or compresses the spring is termed as elastic potential energy. The work done by the applied force against the restoring force of the spring is stored as the elastic potential energy in the spring.

Consider a spring-mass system. Let us assume a mass, m lying on a smooth horizontal table as shown in Figure 4.9. Here, x = 0 is the equilibrium position. One end of the spring is attached to a rigid wall and the other end to the mass.

As long as the spring remains in equilibrium position, its potential energy is zero. Now an external force  _a is applied so that it is stretched by a distance (x) in the direction of the force.

There is a restoring force called spring force  _s developed in the spring which tries to bring the mass back to its original position. This applied force and the spring force are equal in magnitude but opposite in direction i.e., _a =-_s . According Hooke’s law, the restoring force developed in the spring is

The negative sign in the above expression implies that the spring force is always opposite to that of displacement  and k is the force constant. Therefore applied force is _a = + k  . The positive sign implies that the applied force is in the direction of displacement  . The spring force is an example of variable force as it depends on the displacement  . Let the spring be stretched to a small distance d. The work done by the applied force on the spring to stretch it by a displacement  is stored as elastic potential energy.

The applied force a and the displacement d (i.e., here dx ) are in the same direction. As, the initial position is taken as the equilibrium position or mean position, x=0 is the lower limit of integration.

If the initial position is not zero, and if the mass is changed from position x_i to x_{f ,} then the elastic potential energy is

From equations (4.24) and (4.25), we observe that the potential energy of the stretched spring depends on the force constant k and elongation or compression x.

Force-displacement graph for a spring

Since the restoring spring force and displacement are linearly related as F = - k x, and are opposite in direction, the graph between F and x is a straight line with dwelling only in the second and fourth quadrant as shown in Figure 4.10. The elastic potential energy can be easily calculated by drawing a F - x graph. The shaded area (triangle) is the work done by the spring force.

Potential energy-displacement graph for a spring

A compressed or extended spring will transfer its stored potential energy into kinetic energy of the mass attached to the spring. The potential energy-displacement graph is shown in Figure 4.11.

In a frictionless environment, the energy gets transferred from kinetic to potential and potential to kinetic repeatedly such that the total energy of the system remains constant. At the mean position,

The work done on the springs are stored as potential energy in the springs.

k_A>k_B implies that U_B>U_A . Thus, more work is done on B than A.

Conservative and non- conservative forces

Conservative force

A force is said to be a conservative force if the work done by or against the force in moving the body depends only on the initial and final positions of the body and not on the nature of the path followed between the initial and final positions.

Let us consider an object at point A on the Earth. It can be taken to another point B at a height h above the surface of the Earth by three paths as shown in Figure 4.12.

Whatever may be the path, the work done against the gravitational force is the same as long as the initial and final positions are the same. This is the reason why gravitational force is a conservative force. Conservative force is equal to the negative gradient of the potential energy. In one dimensional case,

Examples for conservative forces are elastic spring force, electrostatic force, magnetic force, gravitational force, etc.

Non-conservative force

A force is said to be non-conservative if the work done by or against the force in moving a body depends upon the path between the initial and final positions. This means that the value of work done is different in different paths.

1. Frictional forces are non-conservative forces as the work done against friction depends on the length of the path moved by the body.

2. The force due to air resistance, viscous force are also non-conservative forces as the work done by or against these forces depends upon the velocity of motion.

The properties of conservative and non-conservative forces are summarized in the Table 4.3.

Law of conservation of energy

When an object is thrown upwards its kinetic energy goes on decreasing and consequently its potential energy keeps increasing (neglecting air resistance). When it reaches the highest point its energy is completely potential. Similarly, when the object falls back from a height its kinetic energy increases whereas its potential energy decreases. When it touches the ground its energy is completely kinetic. At the intermediate points the energy is both kinetic and potential as shown in Figure 4.13. When the body reaches the ground the kinetic energy is completely dissipated into some other form of energy like sound, heat, light and deformation of the body etc.

In this example the energy transformation takes place at every point. The sum of kinetic energy and potential energy i.e., the total mechanical energy always remains constant, implying that the total energy is conserved. This is stated as the law of conservation of energy.

The law of conservation of energy states that energy can neither be created nor destroyed. It may be transformed from one form to another but the total energy of an isolated system remains constant.

Figure 4.13 illustrates that, if an object starts from rest at height h, the total energy is purely potential energy (U=mgh) and the kinetic energy (KE) is zero at h. When the object falls at some distance y, the potential energy and the kinetic energy are not zero whereas, the total energy remains same as measured at height h. When the object is about to touch the ground, the potential energy is zero and total energy is purely kinetic.

Motion in a vertical circle

Imagine that a body of mass (m) attached to one end of a massless and inextensible string executes circular motion in a vertical plane with the other end of the string fixed. The length of the string becomes the radius (  ) of the circular path (Figure 4.14).

Let us discuss the motion of the body by taking the free body diagram (FBD) at position where the position vector (  ) makes an angle θ with the vertically downward direction and the instantaneous velocity is as shown in Figure 4.14.

There are two forces acting on the mass.

1.        Gravitational force which acts downward

2.        Tension along the string.

Applying Newton’s second law on the mass, In the tangential direction,

The circle can be divided into four sections A, B, C, D for better understanding of the motion. The four important facts to be understood from the two equations are as follows:

i. The mass is having tangential acceleration (g sin θ) for all values of θ (except θ = 0^o), it is clear that this vertical cirular motion is not a uniform circular motion.

ii. From the equations (4.28) and (4.29) it is understood that as the magnitude of velocity is not a constant in the course of motion, the tension in the string is also not constant.

iii. The  equation T = mg cosθ + mv²/r highlights that in sections A and D of the circle, (for –π/2 < θ < π/2;cosθ is positive)  , the term mg cosθ is always­ greater than zero. Hence the tension cannot vanish even when the velocity vanishes.

iv. The equation mv²/r = T - mg cosθ  further highlights that in sections B and D of the circle, (for –π/2 < θ < 3π/2;cosθ is negative) the second term is  always greater than zero. Hence velocity cannot vanish, even when the tension vanishes.

These points are to be kept in mind while solving problems related to motion in vertical circle.

For the lowest point (1)

When the body is at the lowest point 1, the gravitational force m  which acts on the body (vertically downwards) and another one is the tension ₁ acting vertically upwards, i.e. towards the center. From the equation (4.29), we get

For the highest point (2)

At the highest point 2, both the gravitational force m on the body and the tension ₂ act downwards, i.e. towards the center again.

From equations (4.31) and (4.33), it is understood that T₁ > T₂. The difference in tension T₁ - T₂ is obtained by subtracting equation (4.33) from equation (4.31).

The term [v₁² – v₂²] can be found easily by applying law of conservation of energy at point 1 and also at point 2.

From the law of conservation of energy given in equation (4.35), we get

Minimum speed at the highest point (2)

The body must have a minimum speed at point 2 otherwise, the string will slack before reaching point 2 and the body will not loop the circle. To find this minimum speed let us take the tension T₂ = 0 in equation (4.33).

The body must have a speed at point 2, v2 ≥ √gr  to stay in the circular path.

Minimum speed at the lowest point 1

To have this minimum speed (v₂=√gr)  at point 2, the body must have minimum speed also at point 1.

By making use of equation (4.36) we can find the minimum speed at point 1.

The body must have a speed at point 1, v₁≥√5gr to stay in the circular path.

From equations (4.38) and (4.39), it is clear that the minimum speed at the lowest point 1 should be √5 times more than the minimum speed at the highest point 2, so that the body loops without leaving the circle.

Solved Example Problems for  Kinetic energy

Example 4.7

Two objects of masses 2 kg and 4 kg are moving with the same momentum of 20 kg m s-1.

a. Will they have same kinetic energy?

b. Will they have same speed?

Solution

a. The kinetic energy of the mass is given by

Note that KE1 ≠ KE2 i.e., even though both are having the same momentum, the kinetic energy of both masses is not the same. The kinetic energy of the heavier object has lesser kinetic energy than smaller mass. It is because the kinetic energy is inversely proportional to the mass (KE ∝ 1/m) for a given momentum.

b. As the momentum, p = mv, the two objects will not have same speed.

Solved Example Problems for  Potential Energy

Example 4.8

An object of mass 2 kg is taken to a height 5 m from the ground g = 10ms-2 .

a. Calculate the potential energy stored in the object.

b. Where does this potential energy come from?

c. What external force must act to bring the mass to that height?

d. What is the net force that acts on the object while the object is taken to the height ‘h’?

Solution

a. The potential energy U =m g h = 2 × 10 × 5 = 100 J

Here the positive sign implies that the energy is stored on the mass.

b. This potential energy is transferred from external agency which applies the force on the mass.

c. 

d. From the definition of potential energy, the object must be moved at constant velocity. So the net force acting on the object is zero.

Solved Example Problems for  Elastic Potential Energy

Example 4.9

Let the two springs A and B be such that kA>kB. On which spring will more work has to be done if they are stretched by the same force?

Solution

The work done on the springs are stored as potential energy in the springs.

kA>kB implies that UB>UA . Thus, more work is done on B than A.

Example 4.10

A body of mass m is attached to the spring which is elongated to 25 cm by an applied force from its equilibrium position.

a. Calculate the potential energy stored in the spring-mass system?

b. What is the work done by the spring force in this elongation?

c. Suppose the spring is compressed to the same 25 cm, calculate the potential energy stored and also the work done by the spring force during compression. (The spring constant, k = 0.1 N m-1).

Solution

The spring constant, k = 0.1 N m-1

The displacement, x = 25 cm = 0.25 m

a. The potential energy stored in the spring is given by

Note that the potential energy is defined through the work done by the external agency. The positive sign in the potential energy implies that the energy is transferred from the agency to the object. But the work done by the restoring force in this case is negative since restoring force is in the opposite direction to the displacement direction.

c. During compression also the potential energy stored in the object is the same.

Work done by the restoring spring force during compression is given by

In the case of compression, the restoring spring force acts towards positive x-axis and displacement is along negative x direction.

Solved Example Problems for  Conservative and nonconservative forces

Example 4.11

Compute the work done by the gravitational force for the following cases

Solution

(As the displacement is in two dimension; unit vectors and are used)

a. Since the motion is only vertical, horizontal displacement component dx is zero. Hence, work done by the force along path 1 (of distance h).

Therefore, the total work done by the force along the path 2 is

Note that the work done by the conservative force is independent of the path.

Solved Example Problems for  Law of conservation of energy

Example 4.13

An object of mass 1 kg is falling from the height h = 10 m. Calculate

a. The total energy of an object at h =10 m

b. Potential energy of the object when it is at h = 4 m

c. Kinetic energy of the object when it is at h = 4 m

d. What will be the speed of the object when it hits the ground?

(Assume g =10 m s-2)

Solution

a. The gravitational force is a conservative force. So the total energy remains constant throughout the motion. At h =10 m, the total energyE is entirely potential energy.

b. The potential energy of the object at h = 4 m is

c. Since the total energy is constant throughout the motion, the kinetic energy at h = 4 m must be KE = E - U = 100 - 40 = 60J

Alternatively, the kinetic energy could also be found from velocity of the object at 4 m. At the height 4 m, the object has fallen through a height of 6 m.

The velocity after falling 6 m is calculated from the equation of motion,

d. When the object is just about to hit the ground, the total energy is completely kinetic and the potential energy, U = 0.

Example 4.14

A body of mass 100 kg is lifted to a height 10 m from the ground in two different ways as shown in the figure. What is the work done by the gravity in both the cases? Why is it easier to take the object through a ramp?

Solution

m = 100 kg, h = 10 m

Along path (1):

The minimum force F1 required to move the object to the height of 10 m should be equal to the gravitational force, F1 mg = 100 x 10 = 1000 N

The distance moved along path (1) is, =10 m

The work done on the object along path (1) is

W = Fh = 1000 x 10 = 10,000 J

Along path (2):

In the case of the ramp, the minimum force F2 that we apply on the object to take it up is not equal to mg, it is rather equal to mg sinθ .(mg sin < mg) .

Here, angle θ = 30o

Therefore, F2  = mg sinθ = 100 × 10 ×  sin30o = 100 × 10 × 0.5 = 500N

Hence,  (mg sinθ < mg)

The path covered along the ramp is,

l = h/sin30 = 10/0.5 =20m

The work done on the object along path  (2) is, W = F2 l = 500 × 20 = 10,000 J

Since the gravitational force is a conservative force, the work done by gravity on the object is independent of the path taken.

In both the paths the work done by the gravitational force is 10,000 J

Along path (1): more force needs to be applied against gravity to cover lesser distance .

Along path (2): lesser force needs to be applied against the gravity to cover more distance.

As the force needs to be applied along the ramp is less, it is easier to move the object along the ramp.

Example 4.15

An object of mass m is projected from the ground with initial speed v0.

Find the speed at height h.

Solution

Since the gravitational force is conservative; the total energy is conserved throughout the motion.

Final values of potential energy, kinetic energy and total energy are measured at the height h.

By law of conservation of energy, the initial and final total energies are the same.

Note that in section (2.11.2) similar result is obtained using kinematic equation based on calculus method. However, calculation through energy conservation method is much easier than calculus method.

Example 4.16

An object of mass 2 kg attached to a spring is moved to a distance x =10 m from its equilibrium position. The spring constant k =1 N m-1and assume that the surface is frictionless.

a. When the mass crosses the equilibrium position, what is the speed of the mass?

b. What is the force that acts on the object when the mass crosses the equilibrium position and extremum position x = ± 10 m.

Solution

a. Since the spring force is a conservative force, the total energy is constant. At x =10 m, the total energy is purely potential.

When the mass crosses the equilibrium position x = 0 , the potential energy

The entire energy is purely kinetic energy at this position.

b. Since the restoring spring force is F = - kx, when the object crosses the equilibrium position, it experiences no force. Note that at equilibrium position, the object moves very fast. When the object is at x = +10 m (elongation), the force F = - k x

F = - (1) (10) = - 10 N. Here the negative sign implies that the force is towards equilibrium i.e., towards negative x-axis and when the object is at x = - 10 (compression), it experiences a forces F = - (1) (- 10) = +10 N. Here the positive sign implies that the force points towards positive x-axis.

The object comes to momentary rest at x = ±10m even though it experiences a maximum force at both these points.

Solved Example Problems for  Motion in a vertical circle

Example 4.17

Water in a bucket tied with rope is whirled around in a vertical circle of radius 0.5 m. Calculate the minimum velocity at the lowest point so that the water does not spill from it in the course of motion. (g = 10 ms-2)

Solution