C. Find the Work Done on the Chain by Direct Integration Of

Homework due 5-7-2013

1. A uniform chain of mass M and length l, hangs from a hook in the ceiling. The bottom link is now raised vertically and hung on the hook as shown above on the right.

a. Determine the increase in gravitational potential energy of the chain by considering the change in position of the center of mass of the chain

b Write an equation for the upward external force F(y) required to lift the chain slowly as a function of the vertical distance y,

c. Find the work done on the chain by direct integration of

2. A positive charge distribution exists within a nonconducting spherical region of radius a. The volume charge density p is not uniform but varies with the distance r from the center of the spherical charge distribution, according to the relationship ρ = βr for 0 r a, where β is a positive constant, and ρ=0, for r >a.

a. Show that the total charge Q in the spherical region of radius a is βπa 4

b. In terms of β, r, a, and fundamental constants, determine the magnitude of the electric field at a point a distance r from the center of the spherical charge distribution for each of the following cases. i. r> a ii. r =a iii. 0 < r <a

c. In terms of β, a, and fundamental constants, determine the electric potential at a point a

distance r from the center of the spherical charge distribution for each of the following cases

i. r= a ii. r =0

3. A skier of mass .M is skiing down a frictionless hill that makes an angle θ with the horizontal, as shown in the diagram. The skier starts from rest at time t =0 and is subject to a velocity-dependent drag force due to air resistance of the form F= -bv, where v is the velocity of the skier and b is a positive constant. Express all algebraic answers in terms of M, b, θ, and fundamental constants.

(a). On the dot below that represents the skier, draw a free-body diagram indicating and labeling all of the forces that act on the skier while the skier descends the hill.

(b)  Write a differential equation that can be used to solve for the velocity of the skier as a function of time.

(c)  Determine an expression for the terminal velocity vT, of the skier.

(d)  Solve the differential equation in part (b) to determine the velocity of the skier as a function of time. Showing all your steps.

(e) On the axes below, sketch a graph of the acceleration a of the skier as a function of time t, and indicate the initial value of a. Take downhill as positive.

4. The circuit above contains a capacitor of capacitance C, a power supply of emf ε, two resistors of resistances R1, and R2, and two switches. S1 and S2 . Initially, the capacitor is uncharged and both switches are open. Switch S1, then gets closed at time t = 0.

(a)  Write a differential equation that can be solved to obtain the charge on the capacitor as a function of time t.

(b)  Solve the differential equation in part (a) to determine the charge on the capacitor as a function of time t. Numerical values for the components are given as follows:

ε=12V

C = 0.060 F

R1 = R2 =4700 Ω

(c)  Determine the time at which the capacitor has a voltage 4.0 V across it. After switch S1 has been closed for a long time, switch S2, gets closed at a new time t = 0.

(d) On the axes below, sketch graphs of the current I1, in R1, versus time and of the current I2, in R2, versus time, beginning when switch S2, is closed at new time t = 0. Clearly label which graph is I1, and which is I2.