CBSE Class 11 Physics Work Energy Power Exam Notes

These CBSE Class 12 Physics exam notes on Work, Energy and Power cover work done by constant and variable forces, the sign convention of work, work done by a spring, kinetic and potential energy, conservative and non-conservative forces, the work-energy theorem, conservation of mechanical energy, power, and motion in a vertical circle with tension and minimum speed conditions. They are useful for quick revision before the board exam.

WORK, ENERGY AND POWER

WORK

1. If a constant force F displaces a body through displacement s, the work done W is given by W = F s cos θ = F · s, where s is the magnitude of displacement and θ is the angle between the force and the displacement.
2. The S.I. unit of work is joule, which is the same as newton-metre.
3. If the work is done by a variable force F which displaces a body from position A to B, having position vectors r₁ and r₂, the work W_AB is given by W_AB = ∫ F · ds, where the integral is taken from r₁ to r₂, and ds is a small displacement of the body between A and B.
4. Work is a scalar quantity. If a number of forces are acting, the total work on the body is equal to the sum of the work done by all the forces.

SIGN CONVENTION OF WORK

1. When 0 < θ < 90°, W = F s cos θ is positive. The force supports the motion of the body, so the work done by that force is positive.
2. When π/2 < θ ≤ π, W = F s cos θ is negative. The force opposes the motion of the body, so the work done by that force is negative.

[Figure: Two diagrams. In the first, a block on a horizontal floor is pulled by a force F at an angle θ above the horizontal, and the direction of motion is to the right, so the work is positive. In the second, a block moves to the right while a force F acts at an angle θ greater than 90 degrees, pointing to the left and upward, so the work is negative.]

WORK DONE BY A SPRING

1. If a spring of spring constant k is stretched from initial extension x_i to final extension x_f, the force opposes the extension, so F = -kx.
2. The work dW done in extending the spring through dx is dW = F dx = -kx dx.
3. Total work done: W = ∫ dW = -∫ kx dx, taken from x_i to x_f, which gives W = -(1/2) k (x_f² - x_i²).
4. The work done by a spring force is negative.

Friction also does work. The work done by friction can be negative, zero or positive, depending on whether the friction force is opposing the motion, causing no relative motion between the surfaces, or supporting the motion.

ENERGY

The energy of a body is its capacity to do work. It is a scalar quantity with the same unit as work, which is joule in the S.I. system. In mechanics, kinetic energy and potential energy are involved with the dynamics of the body.

KINETIC ENERGY

Kinetic energy is the energy possessed by a body by virtue of its motion. A body of mass m moving with velocity v has kinetic energy E_k = (1/2) m v².
Since velocity is a relative quantity, kinetic energy is also a relative quantity.

TYPES OF FORCES

Conservative and Non-conservative Forces

1. A force is conservative if the work done by it in moving a particle from one point to another does not depend on the path taken, but depends only on the initial and final positions.
2. The work done by a conservative force around a closed path is zero.
3. Gravitational force, electric force and spring force are examples of conservative forces. All central forces are conservative forces.
4. A force is non-conservative if the work done by it depends on the path followed.
5. The work done by a non-conservative force around a closed path is not zero.
6. Frictional forces and viscous forces are non-conservative. They work in an irreversible manner, and part of the mechanical energy is converted into other forms such as heat and sound.

POTENTIAL ENERGY

Potential energy is the energy of a body possessed by virtue of its position, or the energy possessed by the body due to its state. It is independent of the way in which the body is taken to that state. It is a relative quantity and depends on its value at the reference level.

The change in potential energy is defined as the negative of the work done by the conservative force in carrying a body from the reference position to the position under consideration.

Definition: ΔU = -W_AB, where A is the initial state, B is the final state and W_AB is the work done by the conservative force.

Since potential energy depends only on the work done by a conservative force, it cannot be defined for a non-conservative force, because the work done by such a force depends on the path followed.

RELATIONSHIP BETWEEN FORCE AND POTENTIAL ENERGY

1. If a body is taken from A to B such that there is no net change in kinetic energy, then work done = -change in potential energy.
2. F Δr = U - (U + ΔU) = -ΔU
3. So F_avg = -(ΔU/Δr)
4. As Δr tends to 0, F = -lim (Δr → 0) ΔU/Δr = -∂U/∂r

The force at any point in a conservative field is equal to the rate of change of potential energy at that point, with a negative sign. This relation helps to decide whether an equilibrium point is stable, unstable or neutral.

[Figure: Straight line from A to B, marked with displacement Δr between them. Point A has potential energy U and point B has potential energy U + ΔU.]

WORK-ENERGY THEOREM

The net work done by the resultant force acting on a particle is equal to the change in its kinetic energy. If u and v are the initial and final speeds of a particle of mass m, the net work W_net is given by W_net = (1/2) m v² - (1/2) m u².
If Δk is the change in kinetic energy, then Δk = W_net.

CONSERVATION OF MECHANICAL ENERGY

The sum of kinetic and potential energy in a conservative system is constant, although it can change from one form to another. The sum of the total change in potential energy ΔU and the total change in kinetic energy Δk is zero if only conservative forces act on the system and no energy is lost in overcoming non-conservative forces.

ΔU + Δk = Δ(U + k) = 0
On integrating, U + k = constant.

Thus, if only conservative forces act and do work, the total mechanical energy of the system is conserved. This is called conservation of mechanical energy.

STEPS FOR SOLUTION OF PROBLEMS

1. Draw a sketch of the physical situation for the initial and final positions, by choosing a suitable coordinate system.
2. Specify the reference level for the potential energy and the correct direction of motion of the body or bodies. Any convenient level can be chosen, such as zero gravitational potential energy at ground level. The potential energy of an unstretched spring of natural length is always taken as zero.
3. Write the sum of kinetic energy, gravitational, electrostatic or magnetic potential energy, and spring potential energy in the compressed or stretched state, for the initial and final positions of the body.
4. Equate the sum of the initial energies with the sum of the final energies for conservative forces. Add the contribution of any irreversible frictional forces present, so that the total energy is accounted for in the initial and final states.
5. Solve for the unknown quantity by the process of elimination.

POWER

1. Power is the rate at which work is done. If an amount of work ΔW is done in time Δt, the average power is P_av = ΔW/Δt.
2. Instantaneous power is P = lim (Δt → 0) ΔW/Δt = dW/dt.
3. The work done by a force F on an object with an infinitesimally small displacement ds is dW = F · ds. So the instantaneous power is P = dW/dt = F · ds/dt = F · v.
4. The S.I. unit of power is watt, which is joule per second. Power is a scalar quantity.

MOTION IN A VERTICAL CIRCLE

A particle of mass m is attached to one end of a string and rotated in a vertical circle of radius r with centre O. The speed of the particle decreases as it travels from the lowest point to the highest point, and increases as it travels from the highest point to the lowest point, due to gravity.

If the particle has velocity v at an instant at point A, where the string makes an angle θ with the vertical, the forces acting on it are the tension T directed towards O and the weight mg acting downwards. The net force towards the centre gives the centripetal force:

T - mg cos θ = mv²/r, so T = m (g cos θ + v²/r)

[Figure: Vertical circle of radius r with centre O. The highest point is C at the top, the lowest point is B at the bottom, D and E are points on the left and right at the level of the centre, and A is a point on the circle at angle θ from the vertical. At A the tension acts along the string towards O, the weight mg acts downwards, and its components are mg cos θ along the string and mg sin θ along the tangent.]

If v₀ is the speed of the particle at the highest point, the speed increases as the particle falls through a height h. For a fall from C to A, the vertical height is h = r(1 + cos θ), so v = √(v₀² + 2gh) = √(v₀² + 2gr(1 + cos θ)). Tension does no work.

(i) At the highest point C (θ = 180°):
Tension at C: T_C = m (v₀²/r - g)
The string becomes slack if T_C is negative. So the minimum speed at the highest point is when T_C is just zero, that is, v₀ = √(rg).

(ii) At the lowest point B (θ = 0):
Tension at B: T_B = m (v_B²/r + g), where v_B is the speed at B.
v_B² = v₀² + 4rg = rg + 4rg = 5rg, so v_B = √(5rg)
Minimum tension at B when the particle completes the circle: T_B = m (5rg/r + g) = 6 mg
At point E (θ = 90°): T_E = mv_E²/r
Speed at E: v_E² = v_C² + 2rg = rg + 2rg = 3rg, so v_E = √(3rg)
Tension at E: T_E = m (3rg/r) = 3 mg

(iii) Particle on a circular track (not tied to a string):
If a particle of mass m moves along a circular track of radius r with normal reaction N, and its radius vector makes an angle θ with the vertical, then mg cos θ - N = mv²/r.
At the highest point, mg - N = mv²/r. When N = 0, v = √(rg).
So v = √(rg) is the minimum speed at the highest point with which the particle can move without losing contact with the track.