Homework due 5-2-2013
The circular loop of wire in Figure
1above has a radius of R and carries a
current I. Point P is a distance of R/2
above the center of the loop. Express
algebraic answers to parts (a) and (b) in
terms of R, I and fundamental constants.
(a)
i. State the direction of the magnetic
field B, at point P due to the current in
the loop.
ii. Calculate the magnitude of the magnetic field B, at point P.
A second identical loop also carrying a current I is added at a distance of R above the first
loop as shown in Figure 2 above.
(b) Determine the magnitude of the net magnetic field Bnet, at point P.
A small square loop of wire in which each side has a length s is now placed at point P with
its plane parallel to the plane of each loop, as shown in Figure 3 above. For parts (c) and (d),
assume that the magnetic field between the two circular loops is uniform in the region of the
square loop and has magnitude Bnet.
(c)In terms of Bnet and s. determine the magnetic flux through the square loop.
(d) The square loop is now rotated about an axis in its plane at an angular speed ω. In terms
of Bnet . s. and ω, calculate the induced emf in the loop as a function of time t, assuming that
the loop is horizontal at t=0.
A billiard ball has mass M, radius R, and moment of inertia about the center of mass IC = 2
MR2/5. The ball is struck by a
cue stick along a horizontal line
through the ball's center of mass
so that the ball initially slides
with a velocity vo, as shown
above. As the ball moves across
the rough billiard table (coefficient of sliding friction µk) its motion gradually changes from
pure translation through rolling with slipping to rolling without slipping.
a. Develop an expression for the linear velocity v of the center of the ball as a function of
Homework due 5-2-2013
time while it is rolling with slipping.
b. Develop an expression for the angular velocity ω of the ball as a function of time while
it is rolling with slipping.
c. Determine the time at which the ball begins to roll without slipping.
d. When the ball is struck it acquires an angular momentum about the fixed point P on the
surface of the table. During the subsequent motion the angular momentum about point P
remains constant despite the frictional force. Explain why this is so.
A ring of mass M, radius R, and rotational inertia MR2 is initially sliding on a frictionless surface at
constant velocity Vo to the right, as shown above. At time t = 0 it encounters a surface with coefficient of
friction  and begins sliding and rotating. After traveling a distance L, the ring begins rolling without
sliding. Express all answers to the following in terms of M, R, v0 , , and fundamental constants, as
appropriate.
(a) Starting from Newton’s second law in either translational or rotational form, as appropriate, derive a
differential equation that can be used to solve for the magnitude of the following as the ring is sliding and
rotating.
i. The linear velocity v of the ring as a function of time t
ii. The angular velocity  of the ring as a function of time t
(b) Derive an expression for the magnitude of the following as the ring is sliding and rotating.
i. The linear velocity v of the ring as a function of time t
ii. The angular velocity  of the ring as a function of time t
(c) Derive an expression for the time it takes the ring to travel the distance L.
(d) Derive an expression for the magnitude of the velocity of the ring immediately after it has traveled the
distance L.
(e) Derive an expression for the distance L.