Q- A bullet is shot through two cardboard disks attached a distance D apart to a shaft turning with a rotational period T, as shown. Derive a formula for the bullet speed v in terms of D, T and a measured angle θ between the position of the hole in the first disk and that of the hole in the second. Both of the holes lie at the same radial distance from the shaft.

### Errors: Measuring Speed of Light

Q- Galileo attempted to measure the speed of light by measuring the time elapsed between his opening a lantern and his seeing the return light from his assistant’s lantern, which is opened when the assistant sees the light from Galileo’s lantern. What distance d must separate Galileo from his assistant in order for the human reaction time Δt = 0.2 s, to introduce no more than 15% error in the measured speed of light?

### Harmonic Oscillations

Q- A harmonic force F_{0 }sin(ωt) is applied to mass m_{1} which in turn is coupled to springs of spring constants k_{1, }k_{2} and k_{3}. Let x_{1} and x_{2 }be the deviations from equilibrium for m_{1 }and m_{2}, respectively. The blocks moves on a horizontal frictionless table.

(a) Write the equations of motion for the two blocks. Use x_{1} and x_{2} as your variables.

(b) Find the normal frequencies of the system, that is, when the external force is absent. Assume for this part and the remainder of the problem the springs are the same and the masses are equal.

(c) Solve for the steady state motion with the force, that is find x_{1} and x_{2} as functions of time.

### Radioactivity

Q- Kr-85 is a radioactive isotope that is used in measuring paper thickness. It decays by emission with a half-life of 10.8 years

(a) Write down the decay equation showing the daughter isotope and decay products of this process.

(b) A Kr-85 source had activity 30 mCi when new. Estimate its activity after 5 years' use.

(c) What is the activity of a 30 mCi source in becquerels?

(d) What is the mass of Kr-85 in the source when new?

### Gravitation: Force inversely proportional to cube of distance.

Q- Consider an object of mass m, moving in a circular orbit, subject to a central attractive for F whose magnitude is given by F(r) = h/r^{3}.

(a)What are the dimensions of h?

(b) Show that the angular momentum for the motion is uniquely determined by h and m.

(c) What is the resulting relation between period and radius analogues to Kepler's third law for this Force?

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