z1 + z 2
. 0 = av-z t . When z is linear in t, av-z for the time interval t1 to t2 is
SET UP: From the information given, z (t ) = 6.00 rad/s + (2.00 rad/s 2 )t
EXECUTE: (a) The angular acceleration is positive, sin
IDENTIFY: The upward buoyant force B exerted by the liquid equals the weight of the fluid displaced by the
object. Since the object floats the buoyant force equals its weight.
SET UP: Glycerin has density gly = 1.26 103 kg/m 3 and seawater has densi
IDENTIFY: Apply = m / V to relate the densities and volumes for the two spheres.
For a sphere, V = 4 r 3 . For lead, l = 11.3 103 kg/m 3 and for aluminum, a = 2.7 103 kg/m3 .
= 1.6 .
IDENTIFY: Follow the procedure specified in the hint.
SET UP: Denote the position of a piece of the spring by l ; l = 0 is the fixed point and l = L is the moving end of the
spring. Then the velocity of the point corresponding to l , denoted u ,
The object oscillates as a physical pendulum, so f =
. Use the parallel-axis theorem,
I = I cm + Md 2 , to find the moment of inertia of each stick about an axis at the hook.
SET UP: The center of mass of the square object
IDENTIFY and SET UP: The amplitude is the maximum displacement from equilibrium. In one period the object
goes from x = + A to x = A and returns.
EXECUTE: (a) A = 0.120 m
(b) 0.800 s = T / 2 so the period is 1.60 s
(c) f = = 0.625 Hz
F = ma
to the motion of the baseball. v =
rD = 6 103 m .
(a) Fg = marad gives G
(6.673 1011 N m 2 /kg 2 )(2.0 1015 kg)
=m . v=
= 4.7 m/s
6 103 m
4.7 m/s = 11 mph , which is easy to a
IDENTIFY: Apply Eq.(12.9) to the particle-earth and particle-moon systems.
SET UP: When the particle is a distance r from the center of the earth, it is a distance REM r from the center of
EXECUTE: (a) The total gravitational potenti
IDENTIFY: Use Eq.(12.2) to find the force each point mass exerts on the particle, find the net force, and use
Newtons second law to calculate the acceleration.
SET UP: Each force is attractive. The particle (mass m) is a distance r1 = 0.200 m from m1
IDENTIFY: The center of gravity of the object must have the same x coordinate as the hook. Use Eq.(11.3) for
xcm . The mass of a segment is proportional to its length. Define to be the mass per unit length, so mi = li ,
where li is the length of a p
IDENTIFY: Apply the first and second conditions of equilibrium to the shelf.
SET UP: The free-body diagram for the shelf is given in Figure 11.8. Take the axis at the left-hand end of the
shelf and let counterclockwise torque be positive. The center
IDENTIFY: The precession angular velocity is =
, where is in rad/s. Also apply
F = ma to the
SET UP: The total mass of the gyroscope is mr + mf = 0.140 kg + 0.0250 kg = 0.165 kg .
2 rad 2 rad
= 2.856 rad/s .
IDENTIFY: Apply = FR and P = .
SET UP: 1 hp = 746 W . rad/s = 30 rev/min
EXECUTE: (a) With no load, the only torque to be overcome is friction in the bearings (neglecting air friction),
and the bearing radius is small compared to the blade radius,
IDENTIFY: Use = Fl = rF sin to calculate the magnitude of each torque and use the right-hand rule to
determine the direction of each torque. Add the torques to find the net torque.
SET UP: Let counterclockwise torques be positive. For the 11.9 N forc
IDENTIFY: Use the equations in Table 9.2. I for the rod is the sum of I for each segment. The parallel-axis
theorem says I p = I cm + Md 2 .
SET UP: The bent rod and axes a and b are shown in Figure 9.59. Each segment has length L / 2 and mass M / 2
IDENTIFY: Apply Bernoullis equation and the equation of continuity.
SET UP: Example 14.8 says the speed of efflux is
2 gh , where h is the distance of the hole below the surface of
EXECUTE: (a) v3 A3 = 2 g ( y1 y3 ) A3 = 2(9.80 m/s 2 )(8.