Physics 105A
Analytical Mechanics
Midterm Review
28 June 2016
Manuel Calderón de la Barca Sánchez
Leaving the sphere
 A small mass rests on top of a fixed frictionless sphere.
The mass is given a tiny kick and slides downward under
gravity. At what point does it lose contact with the
sphere?
– There are at least two ways to do this.
28 June 2016
MCBS
Rotating Hoop
 A bead lies on a frictionless hoop of radius R that rotates
around a vertical diameter with constant angular
frequency ω. What should ω be so that the bead
maintains the same position on the hoop, at an angle θ
with respect to the vertical?
 There is a special value of ω; what is it, and why is it
special?
28 June 2016
MCBS
Atwood’s 4
 Consider the Atwood’s machine shown (or look at the
cover of your text).
 If the number of pulleys that have string passing beneath
them is N instead of 3, find the acceleration of the
masses.
28 June 2016
MCBS
Cart, ball and tube
 A cart is held at rest on an inclined plane. A tube is
positioned in the cart with its axis perpendicular to the
plane. The cart is released, and at some later time a ball
is fired from the tube. Will the ball eventually land back in
the tube?
28 June 2016
MCBS
Car on a banked track
 A car travels around a circular banked track of radius R.
 The angle of the bank is q, and the coefficient of static
friction between the tires and the track is m.
 For what range of speeds does the car not slip?
Mercedes Benz
AMG SLS Roadster
28 June 2016
MCBS
Heading to zero
 A particle moves toward x=0 under the influence of a
potential of the form
V ( x)   A | x |
n
 where A>0 and n>0.
The particle has barely enough
energy to reach x=0. For what values of n will it reach
x=0 in a finite time?
28 June 2016
MCBS
Line of Pulleys
 N+2 equal masses hang from a system of pulleys as
shown. What are the accelerations of all the masses?
N masses
28 June 2016
MCBS