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Part A
1a. C (apply conservation of momentum to find: v_final
= (2/3) v_0)
1b. B (calculate initial kinetic energy (M * v_0^2)
and final kinetic energy ((2/3)M * v_0^2), and find that
K changes: -(1/3)M * v_0^2 is lost kinetic energy,
therefore inelastic)
2a. B (impulse = change in momentum for baseball, so ball's
v_final = 68 m/s)
2b. F (average force = impulse / time = 24,000 N)
3a. E (frequency = 7200 rev/min = 120 rev/s; angular frequency
= 754 rad/s; angular accel = DeltaOmega/DeltaTime = -3000 rad/s^2)
3b. D (use rotational kinematics: solve for DeltaTheta
in either: w^2 = w_0^2 + 2*alpha*DeltaTheta OR DeltaTheta =
w_0*t + (1/2)alpha*t^2; then convert 94 rad into 15 rev)
4. D (either suspend at the center of mass of the two masses
(0.29 L away from left-hand mass); OR find pivot point
location on rod about which torques sum to zero (0.29 L))
5a. A (omega = 2*pi / period)
5b. E (use right-hand rule)
6. C (total angular momentum of Earth plus all passengers
must remain constant: if the net passengers' eastward omega increases,
Earth's eastward omega must decrease)
7. D (A through C are all true; a small rate of
precession (or zero precession) is most desirable for navigational
gyroscopes)
8. A (as angular speed increases: moment of inertia is
unchanged; torque is present only is angular speed is changing;
and precession (if any) diminishes)
Part B
1a. 97 N*m (torque = r * (m*g)
* sin(theta) = 0.18m * 55kg * 9.81m/s^2 * sin(90deg))
1b. D (same formula as part (a), but theta = 45deg)
1c. A (same formula as part (a), but theta = 0deg)
1d. A (when r is maximized, torque is maximized)
1e. B (the same length of chain that wraps on the pedal
gear per second = length of chain that unwraps from rear-wheel
gear per second: w_p * R_p = w_1 * R_1)
2a. v_B = sqrt(2 * g * R) (using conservation
of energy: U_grav_init = K_final)
2b. v_Bf = sqrt(g * R / 2) (using conservation
of momentum in collision: M * v_Binit = (2M) * v_Bfinal )
2c. DeltaK = -(1/2) M * g * R
(calculate total K before and after collision, where K
= (1/2) M * v^2 )
2d. d = R / (4 * mu_k) (using either
conservation of energy or kinematics)