Assignment 1

Problem 1:

An elevator car of maximum mass mass m0 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 σy, (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/s2). 
Assume that m0 = 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 σy for the case of a steel cable. 
Use σ ysteel = 1380 MPa and ρsteel = 7.7 g/cm3.

Problem 2:

A solid iron cylinder (density = 7.87 g/cm3) of radius r = 5 cm and length l = 20 cm rolls down a ramp which has an incline of 20o (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?

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Problem 3:

imageConsider the circuit shown.  At t = 0 the switch is closed.
(a)  What are the values of the currents through the resistors R1, R2, and R3 just after the switch has been closed and a long time after the switch has been closed?
(b)  Find the currents through the resistors R1, R2, and R3 as a function of time for t > 0.

Problem 4:

Consider a series RLC circuit shown below. The circuit is driven by a sinusoidal voltage that has amplitude Vin.

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(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 ωLC = 1/(LC)½ and ωRC = 1/RC.

Problem 5:

A switch is closed to charge a capacitor C from a battery of voltage Vo through a resistance R.  (Fig. a).  The capacitor is a circular parallel plate capacitor of area A = πb2, 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?

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(a)                                                   (b)