Revision Note 7: Momentum and Collisions

Momentum:

p = mv

What is important is change in momentum,

F∆t = ∆p = m∆v

Impulse:

F∆t is called the impulse, which is equal to change in momentum.

Use calculus when force varies (generally with time):

Center of mass:

For n different masses, the center of mass (x,y) is:

Problem Solving Tips:

Tip 7.1:

For solving momentum problems, first resolve momentum into two orthogonal axes. Then:

p_i,x = p_f,x

p_i,y = p_f,y

The above apply to all types of momentum problems.

The issue is with the number of unknowns and number of equations. There are now three types:

(a) Apply the principle of conservation of momentum only. Some parts of the final momentum are given – maybe the angle or the magnitude – to provide the extra equations to enable solution.

(b) Completely Elastic. No energy lost.  In this case we have the extra equation:

KE_i = KE_f

Or if springs are involved:

KE_i + PE_i = KE_f + PE_f

(c ) Completely inelastic. No, not all energy is lost, but some definitely is lost. The definition of a completely inelastic collision is that the two objects stick together after collision, so their final velocities are the same:

v_1,f = v_2,f

Broken down into two components:

v_x,1,f = v_x,2,f

v_y,1,f = v_y,2,f

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.

Revision Note 5: Friction

Frictional force (more correctly, the maximum frictional force) is normal force times coefficient of static or dynamic friction (as the case may be).

F_fr = μN

The normal force (N) that is used to compute frictional force is best computed by considering the force by the surface on the body – in other words you need the free-body diagram of the body. This is the best method to compute the force by the body on the surface, as by Newton’s third law the two forces are numerically the same.

Problem Solving Tips:

Tip 5.1:

What is F_fr  for a ramp of incline θ to horizontal when the coefficient of friction is μ? By a previous tip (Tip 2.3), N = mg.cosθ. So,

F_fr,θ = μN = μ mg.cosθ

Tip 5.2: Up-down intuition:

Relating θ with normal force, N, and frictional force, F_fr,θ:

θ ↑        N ↓      F_fr,θ  ↓

The effect of friction decreases as the ramp becomes more and more vertical.