An elevator car of maximum mass mass m_{0} when fully loaded is connected to a counter
balance of the same mass by a cable of length L.

(a) Derive an expression for the cross sectional area A of the cable such
that the yield strength σ_{y},
(a stress, force/area) of the cable is not exceeded.

(b) Let ρ be the mass density (mass per unit volume) of the cable and the acceleration due to gravity (9.8 m/s^{2}).

Assume that m_{0} = 8800 kg and the elevator car carries its load up a
skyscraper of height 5 km (!).

Calculate the minimum cross sectional area of
the
cable without exceeding σ_{y} for the case of a steel cable.

Use σ_{ ysteel}
= 1380 MPa and ρ_{steel} = 7.7 g/cm^{3}.

A solid iron cylinder (density = 7.87 g/cm^{3}) of
radius r = 5 cm and length l = 20 cm rolls down a ramp which has an incline of
20^{o} (no sliding). The initial height is 3 m above ground.

(a) What is the magnitude of the linear acceleration at half the height?

(b)
The cylinder arrives at the bottom of the ramp. What is the angular
momentum of the cylinder about its central axis if it suddenly lifted up from
the ground at the two ends of this axis?

Consider the circuit shown. At t = 0 the switch is closed.

(a) What are the values of the currents through the resistors R_{1}, R_{2},
and R_{3} just after the switch has been closed and a long time after
the switch has been closed?

(b) Find the currents through the resistors R_{1}, R_{2}, and R_{3}
as a function of time for t > 0.

Consider a series RLC circuit shown below. The circuit is driven by a
sinusoidal voltage that has amplitude V_{in}.

(a) At what input frequency is the voltage amplitude between points A and B
maximum (assuming a maximum exist)?

(b) For a low damping case determine the voltage
amplitude between points A and B at the resonance frequency.

(c) What is the phase shift in this resonant case?

(d) What is the average power dissipated by the circuit (that is by a series
connection of R, L and C) at the resonance?

Hint: To simplify the notation define ω_{LC} = 1/(LC)^{½} and ω_{RC} = 1/RC.

A switch is closed to charge a capacitor C from a battery
of voltage V_{o }through a resistance R. (Fig. a). The capacitor is a
circular parallel plate capacitor of area A =
πb^{2}, plate spacing d << b and dielectric constant
ε. (Fig. b) (Neglect fringing fields
and retardation effects.)

(a) Calculate the (time dependent) charge on the capacitor
and the current through the resistor.

(b) What is the electric field inside the capacitor?

(c) What is the magnetic field inside the capacitor?

(a) (b)