Unformatted text preview: F = C1 rv + C2r2 v2 F = mM G
r2 F m1r1 = m2 r2 2( + r 3
T 2 = 4 (mr1 + m2 ))
= dp = ma dW = F � dr acent = v = !2r
U = �mM G
K = 1 mv2
K = 1 I !2
U = mgh
U = 1 kx2
Etot = K + U = 2 mv2 � mM G = �mM G
I = mi ri2
i v = !r
dt dL = r � F = I = dt
q ! = g=l 1 !pr = L s q ! = dt
d 2 T = 2!
L = I!
T 2 = 4GM I ! = k=m
0 F dt = pf � pi Solid disk of mass M and radius R rotating about its cylindrical axis: I = 1 M R2
Solid sphere of mass M and radius R rotating about an axis through its center: I = 2 M R2
5 vf � vi = �u ln mf � gt
f = f 1 + v cos
c I = Icm + M d2 0 P V = nRT
p R = 8:31J=K
E = hf = hc = 1 � v cos
c dP = � g
k = 1:38 � 10 23 J=K NA = 6:02 � 1023 P V = N kT � L= L T V= V T 34 � J � sec h = 2h = 1:05 � 10 h = 6:6 � 10 0 Iz = Ix + Iy 1 v2 + gy + P = 1 v2 + gy + P
21 34 � J � sec Problem 1 (16 points) A gunner res a bullet of mass m with speed v0 at an angle from the horizontal plane.
Assume the bullet is red from ground level. The gravitational acceleration is g. Ignore air
drag. Express your answers in terms of m, g, v0 , and .
d. (4) When does the bullet reach its highest point?
How high is that above the ground?
With what speed will the bullet hit the ground?
What is the horizontal distance that the bullet has traveled when it hits the ground? Problem 2 (15 points) A pendulum has length l (the string is \massless"). The bob has a mass of m. We release
the bob with zero speed when the string makes an angle = 90� with the vertical. Friction
of any kind can be ignored. The gravitational acceleration is g. Express your answers in
terms of l, m, and g.
a. (5) What is the speed of the bob when it reaches its lowest point ( = 0�)?
b. (5) What is the tension in the string when = 0�?
c. (5) How much work was done by gravity and how much by the tension in the string
between the moment of release and the moment that the bob reaches its lowest
point? Problem 3 (24 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. (6)
d. (6) 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 4 (20 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. (4) When the displacement angle is , what then is the torque relative to point P ?
b. (4) What is the moment of inertia for rotation about the axis through P ?
c. (4) 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, �max, is very small, and
the motion is a near perfect simple harmonic oscillation.
d. (4) What is the period of oscillation?
e. (4) As the disk oscillates, is there any force that the axis at P exerts on the disk? Explain
your answer. Problem 5 (25 points) An apple of mass m is swung around on a string in a circle in a horizontal plane with a
constant speed. The string makes an angle with the vertical (see the gure). The radius
of the circle is R it takes sec for the apple to make one complete rotation the direction
of rotation is indicated in the gure the apple, at S, is coming toward you. Q is the center
of the circle. QP is vertical. SQ is in the +x-direction, QP in the +y-direction, and the +zdirection is tangent to the circle at S and points toward you. The gravitational acceleration
is g assume that the string is massless. Express your answers in terms of m, R, , and g.
z S Q
R a. (4) Make a free body diagram for the apple at S.
b. (4) What is the velocity of the apple at S (magnitude and direction), and what is its
angular velocity !?
c. (4) What is the apple's centripetal acceleration at S (magnitude and direction)?
d. (4) At S, what is the direction and magnitude of the sum of all forces acting on the
apple. Indicate this one force in the gure or in a separate sketch, and mark this
e. (9) What is ? Problem 6 (25 points) A particle of mass m1 and speed v1 (in the +x-direction) collides with another particle of
mass m2 . m2 is at rest before the collision occurs, thus v2=0. After the collision, the particles
have velocities v1 and v2 in the x-y plane in the directions of 1 and 2 with the x-axis (see
gure). There are no external forces. Express all your answers in terms of m1 , m2 , v1 , 1 ,
and 2 .
0 0 ’
v1 x a. (3)
e. (8) m2 m1 y v1 v2 = 0 m1 θ1 m2 θ2
v2 What is the total momentum before the collision (direction and magnitude)?
What is the total momentum after the collision (direction and magnitude)?
What is the total kinetic energy before the collision?
What is the ratio of the speeds v2 =v1?
What is the magnitude (speed) of v1?
0 0 0 Problem 7 (25 points) Imagine a spherical, non-rotating planet of mass M and radius R. The planet has no
atmosphere. A spacecraft of mass m (m � M ), is launched from the surface of the planet
with speed v0 at an angle of 30� to the local vertical. The rocket burn is very short. Thus you
may assume that when the spacecraft has a speed v0 , it has not yet moved any appreciable
a. (4) The speed v0 is so high that the orbit is not bound. What is the minimum speed for
which this is the case?
Now imagine that the orbit is bound and that in its subsequent orbit the spacecraft reaches
a maximum distance of 15R from the center of the planet. At this distance the speed is V .
e. (8) What is the ratio of v0 =V ?
What is the total energy of the spacecraft immediately after launch?
What is the total energy of the spacecraft when it is farthest away from the planet?
Write down one equation which would allow you to solve for v0 in terms of M , G,
and R (we are not asking you to solve this equation). Problem 8 (18 points) A cylindrical container of length L is full to the brim with a liquid which has mass density
. It is placed on a weigh-scale the scale reading is W . A light ball which would oat on the
liquid if allowed to do so, of volume V and mass m is pushed gently down and held beneath
the surface of the liquid with a rigid rod of negligible volume as shown on the left.
rigid rod L a. (4) What is the mass M of liquid which over owed while the ball was being pushed into
b. (6) What is the reading of the scale when the ball is fully immersed? Give your reasons.
c. (8) If instead of being pushed down by a rod, the ball is held in place by a thin string
attached to the bottom of the container as shown on the right. What is the tension
T in the string, and what is the reading on the scale? Problem 9 (15 points) Show that if the temperature T in the atmosphere were independent of altitude, the pressure
p as a function of altitude h would be
mgh p = p0 e� kT where m is the average mass of an air molecule, and p0 is the pressure at sea level. Problem 10 (17 points) A bowling ball of mass m and radius R sits on the smooth oor of a subway car. If the car
has a horizontal acceleration a1 , what is the acceleration a2 of the ball? Assume that the
ball rolls without slipping. The gravitational acceleration is g. ...
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