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Here is exam2: exam2.doc

Problems 2, 3 and 4 are actually problems 10.92, 12.86 and 13.54 in the Problem Supplement Book and here are the solutions:

• p2.pdf (also see below)
• p3.pdf
• p4.pdf

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Problem 1:

Just to remind you - this is how you calculate the moment of inertia of a solid sphere of uniform density:

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Part (a):

Part (b):

In the above drawing the mass of the differential volume will be somewhat modified since the density now depends on r. So the total mass of the sphere is given by:

so from this we have:

Part (c):

Again, refering to the above drawing and replacing the density by one that is linear in r:

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Problem 2

I will label the initial velocity as v₁ and the final velocities as u₁ and u₂. I will also assume that m₂=2m₁. Conservation of momentum yields:

Now user v₁ = 10 m/s and 30^o for the angle of the u₁ vector. This becomes:

Squaring and adding yields:

Equation 1

The plot of u₂ vs u₁ is shown below as part of the Mathematica output. Note that the max velocity for particle 1 corresponds to the requirement of elastic scattering. If we assume elastic scattering then:

Use this in Eq 1 to get a quadratic:

The solutions are shown below - we choose the positive root yielding 9.34 m/s as the max vel for particle 1.