Revision Note 6: Work and Energy

Work done

Work done by a single force, F, which applies through a displacement, d, is given by the scalar product:

W = F.d = Fd cosθ

You need calculus if any one of the following is true:

·        The magnitude of the force varies (generally as a function of distance)

·        The θ varies (generally as a function of distance)

In the Calculus version, add up the work done by the force for small distances (dx) – which requires integration from x = x_i = initial position to x = x_f = final position. In other words:

If multiple forces act on the object then the total work done by all forces together is:

·        Algebraic sum of the work done by each force individually.

·        Work done by the net force.

·        Sum of work done over small distances covering the whole distance (integration method)

Work and Energy

If A applies a force on B and does W work, then B does  –W work on A. This comes from Newton’s third law. The object A also transfers W amount of energy to B, and the object B transfers  –W of energy to A.

Work energy theorem:

W_{all forces} = ∆K

Potential energy theorem:

W_grav = -∆U

If the only force is gravitation (or any conservative force) then W_grav = -∆U = ∆K. In this case,

∆K + ∆U = 0

or K + U = constant, which means for initial and final positions:

K_i + U_i = K_f + U_f

The intuition of conservative forces is that work doesn’t get lost and only gets transferred between two bodies – hence talking about potential energy makes sense. PE won’t make sense in case energy leaks out.

Potential Energy

A natural way to measure gravitation potential energy is to set the gravitational PE to 0 when the separation of bodies is infinity. In that case:

Similar considerations apply to energy of an electron around the nucleus and thus that electron while in orbit has negative energy.

Potential energy of a spring of spring constant k compressed by x is given by:

Finding F_grav(x) from U(x):

Power

If P is the power (scalar), then for a constant force, F:

Problem Solving Tips:

Tip 6.1:

For computing W, since cosθ = cos(-θ) we don’t care if the angle is measured from Force to displacement or vice-versa – we just say that it is the angle between the Force and displacement vectors.

However the work done can be negative or zero – see the following diagram:

Scan It

Tip 6.2:

For work, what is important is the displacement, not the distance (though see Work done by friction below). So work done by a constant force through a round trip is zero since displacement is 0.

Tip 6.3:

For work done by friction, the magnitude of the force F_fr = μN continually adjusts to oppose the motion, so θ=180 always.  So for each dx the work done is W_fr = -F_fr.dx. Total work is W_fr = -F_fr.D where D = distance covered. In this case work depends on distance rather than displacement.

Tip 6.4:

When conservative forces are acting, the work done doesn’t depend on path. The potential energy is dependent on location, not the path taken to it.

Tip 6.5:

The spring has the same potential energy for the same amount of compression or same amount of extension. This is because (x)² = (-x)².

Tip 6.6:

Work done by conservative forces = difference in the potential energy between starting and ending locations.