Unformatted text preview: F = C1 rv + C2r2 v2 F = mM G
U = ;mM G
r U = mgh F = dp = ma dW = F dr ac
K = 1 mv2
2 L=r p U = 1 kx2
Etot = K + U = 2 mv2 ; mM G = ;mM G
2a m1r1 = m2 r2 q ! = g=l 1 = d!
dt � I= � 2( + r 3
T 2 = 4 (mr1 + m2 ))
G v = !r = r F = I = dL
dt v = !2r
K = 1 I !2
2 � !pr = L s ! = dt
d 2 T = 2!
L = I!
T 2 = 4GM I= 2 X
i mi ri2 q ! = k=m
0 t F dt = pf ; pi Solid disk of mass M and radius R rotating about its cylindrical axis: I = 1 M R2
2 vf ; vi = ;u ln mf ; gt
f = f 1 + v cos
0 0 I = Icm + M d2
1 ; v cos
c Iz = Ix + Iy Problem 1 (35 points) An unknown mass, m1 , hangs from a massless string and descends with an acceleration g=2.
The other end is attached to a mass m2 which slides on a frictionless horizontal table. The
string goes over a uniform cylinder of mass m2 =2 and radius R (see gure). The cylinder
rotates about a horizontal axis without friction and the string does not slip on the cylinder.
Express your answers in parts b, c, and d in terms of g, m2, and R. m2
R a = g/2
m1 a. (8)
d. (9) Draw free-body diagrams for the cylinder and the two masses.
What is the tension in the horizontal section of the string?
What is the tension in the vertical section of the string?
What is the value of the unknown mass m1 ? Problem 2 (30 points) A bullet of mass m1 is red into a pendulum of mass m2 and length L. The speed of the
bullet as it enters the mass m2 is V1 (see gure). m1 V
1 m2 First, assume that the collision is elastic, and that m1 � m2.
a. (6) If the pendulum is initially at rest, what is the speed of the bullet after the collision?
b. (8) Now suppose that when the collision occurs, the pendulum, at the bottom of its swing,
is moving to the left with velocity V2. What now is the speed of the bullet after the
elastic collision? Now assume that the collision is completely inelastic. The pendulum is at rest
before the collision, m1 < m2, but the speed V1 of the bullet is unknown.
c. (8) After the collision the pendulum moves to the right and it comes to a halt when the
string makes an angle max with the vertical. What was the speed of the bullet?
Substitute in your answer max = 0. Does your result make sense?
d. (8) Could max be 90 ? Explain your answer. � Problem 3 (35 points) A solid, uniform disk of mass M and radius R is oscillating about an axis through P . The
axis is perpendicular to the plane of the disk. Friction at P is negligibly small and can be
ignored. The distance from P to the center, C, of the disk is b (see gure). The gravitational
acceleration is g. P
C R θ a. (7) When the displacement angle is , what then is the torque relative to point P ?
b. (7) What is the moment of inertia for rotation about the axis through P ?
c. (7) The torque causes an angular acceleration about the axis through P . Write down the
equation of motion in terms of the angle and the angular acceleration. As the disk oscillates, the maximum displacement angle,
the motion is a near perfect simple harmonic oscillation. max, is very small, and d. (7) What is the period of oscillation?
e. (7) As the disk oscillates, is there any force that the axis at P exerts on the disk? Explain
your answer. ...
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