5 Work, Energy and Power

Question 1

Define work done, its S.I. unit. State the formula to calculate work done and whether it is a scalar or vector quantity.

Answer 1

Work done by a constant force is the product of the force (F) and the displacement (s*cos theta) in the direction of the force. (It is a means of transferring energy to change the mechanical energy of a system.) It is measured in joules J and is a scalar quantity as the dot product of two vectors (Force and displacement) gives a scalar quantity. Work done = Fscostheta, where theta is the angle subtended by two outwardly pointing angles.

Question 2

Define work done by a variable force on the system.

Answer 2

Work done by a variable force on the system is equal to the area under the force-displacement graph.

Question 3

Define work done by an external force on a spring.

Answer 3

The external force F needed to produce an extension or compression x in a spring that obeys Hooke’s Law is F = kx, where k is the spring constant. This force is a variable force as its magnitude depends on the extension or compression of the spring.

Work done by an external force on a spring to stretch an unextended spring by x is equal to the area under the force-extension graph = 1/2Fx = 1/2kx^2

(NOT W = Fscostheta as it is a varying force and the force increases proportionally with extension, hence work done per unit of extension is increasing.)

Question 4

Define the limit of proportionality.

Answer 4

Once exceeded, the extension of the spring no longer increases proportionally with force and instead the spring has gotten less stiff and a smaller and smaller amount of force is needed to extend the spring.

Question 5

Define the limit of elasticity.

Answer 5

It is point where the spring is permanently deformed and no longer extends.

Question 6

Define the work done by a gas.

Answer 6

Work done by a gas is the work done by force F in displacing the piston of cross-sectional area A through a small distance x. It is represented by the formula Wgas = F(change in x) = constant pressure of gasA(change in x) = constant pressure of gas*volume moved.

Question 7

Define chemical potential energy.

Answer 7

Potential energy related to the structural arrangement of atoms or molecules in a substance.

Question 8

Define electrical energy/electrostatic potential energy.

Answer 8

The energy possessed by charge carriers moving under the influence of potential difference. / The energy due to the position of a charge in an electric field of another charge/configuration of charges. (Note that it belongs to the system of the charges and not to any charge alone.)

Question 9

Define internal energy.

Answer 9

Sum of the microscopic kinetic energies (due to random motion of atoms and molecules) and potential energies (due to intermolecular forces)

Question 10

Define gravitational potential energy. State the formula used to calculate Ep/U.

Answer 10

Energy due to the position of a mass in a gravitational field. (Note that it belongs to the system of the two systems and not to either mass alone.)

Consider an object being raised upwards at constant velocity from a height of h1 to h2 near the earth’s surface where the gravitational field is assumed to be constant. Since the velocity at which the object moves is constant, the force F required to lift the object must be equal to its weight.

Work done in displacing the centre of mass of an object vertically upward/Gain in Ep. of the object = Fscos0 (F and s are in the same direction) = mg(h2-h1) = mgh (assuming h1 is the reference level - can be assigned to any convenient point)

Question 11

Define elastic potential energy. State the formula used to calculate Ue.

Answer 11

Energy stored in an object which has had its shape changed elastically (stretching or compressing).

Ue = 1/2kx^2

Note that the Ue of an unstretched object is 0 unlike in U where the point of zero energy can be arbitrarily chosen.

Question 12

Define kinetic energy. State the formula used to calculate K.E.

Answer 12

Energy due to the motion of the body. Consider a body of mass m moving with an initial velocity u that is acted upon by a constant resultant force that is parallel to u. The body accelerates with uniform acceleration to a final velocity v over displacement s.

Ek. = 1/2mv^2 is the translation K.E of an object moving along a path and is always positive.

Question 13

Define mechanical energy. When is mechanical energy conserved? State and explain the energy equation of an isolated and non-isolated system.

Answer 13

K.E. + P.E

Mechanical energy is conserved when the resultant external force acting on the system is zero and the system becomes an isolated system where the energy equation of the system is (Ep + Ek)initial = (Ep + Ek)final.

However, in a non-isolated system where the is work done by an external force, the energy equation of the system becomes (Ep + Ek)initial + Wf = (Ep + Ek). When work done is positive (object undergoing acceleration), the total mechanical energy of the system increases. When work done is negative (object experiences friction and slows down), the total mechanical energy of the system decreases.

Question 14

State the relationship between force and potential energy. What can you infer from the relationship?

Answer 14

For a field of force, the relationship between force F and the potential energy U for one-dimensional motion is given by F = -dU/dx. Hence, the force acts in the opposite direction of displacement which has resulted in an increase in potential energy U.

We can infer that the magnitude of the force at point x is equal to the gradient of the potential energy curve at x (potential energy - displacement graph) and the direction of the force is the direction for decreasing potential energy.

Question 15

Define the law of conservation of energy.

Answer 15

The law of conservation of energy states that energy cannot be created or destroyed, it can only be converted from one form to another.

Question 16

Define efficiency. What makes up the useful energy output?

Answer 16

Efficiency gives a measure of how much of the total energy may be considered useful and it not ‘lost’. It is given by useful energy output/total energy input *100%. The useful energy output is made up of energy that is used to fulfil the object’s purpose. The energy lost can be the heat and sound that is generated, energy absorbed by the apparatus, transferred to the environment, heat generated in cables.

Question 17

Define power. State the equation used to calculate power, average power and its S.I. unit.

Answer 17

Power is the rate of work done or energy conversion with respect to time. It is measured in watt w.

Power P = dW/dt.

Average power P = Work done in time t/time t.

Question 18

State the relationship between power, force and velocity.

Answer 18

If a constant force F is applied and does work by moving its point of application a displacement s in time t, then the power supplied is given by P = dW/dt = d(Fs) /dt = Fds/dt = Fv where v is the velocity of the point of application.

Question 19

Derive the formula used to calculate the mass of air swept through the area swept out by the blades of a turbine per second.

Answer 19

Mass per second = dm/dt = dpv/dt = p*dAx/dt =pAv where x is the length of the air column passing through the blades hence Av is the volume flow rate.

Question 20

Explain the relationship between the power input of a cyclist travelling up a slope at constant speed, the angle of the slope, resistive forces and work done.

Answer 20

At constant speed, the power input by the cyclist is equal to the sum of the rate at which work is done against the resistive force and the rate of increase in gravitational potential energy:

Power input by cyclist = Power dissipated by resistive force + rate of increase of GPE

Power input = fv + mgh/t = fv +mgdsintheta/t = fv + mgvsintheta