A basketball (which can be closely modeled as a hollow spherical shell) rolls down a mountainside into a valley and then up the opposite side, starting from rest at a height H0 above the bottom. In the figure, the rough part of the terrain prevents slipping while the smooth part has no friction.
a) How high, in terms of H0, will it go up the other side?
b) Why doesn't the ball return to height H0? Has it lost any of its original potential energy?
A string is wrapped several times around the rim of a small hoop with radius 8.00 cm and mass 0.180 kg. The free end of the string is held in place and the hoop is released from rest (the figure). After the hoop has descended 75.0 cm , calculate
a) the angular speed of the rotating hoop and
b) the speed of its center
Biomedical measurements show that the arms and hands together typically make up 13.0 % of a persons mass, while the legs and feet together account for 37.0 % . For a rough (but reasonable) calculation, we can model the arms and legs as thin uniform bars pivoting about the shoulder and hip, respectively. Let us consider a 76.0 kg person having arms 68.0 cm long and legs 93.0 cm long. The person is running at 12.0 km/h , with his arms and legs each swinging through 30° in 1/2 s. Assume that the arms and legs are kept straight.
a) What is the average angular velocity of his arms and legs?
b) Calculate the amount of rotational kinetic energy in this persons arms and legs as he walks.
c) What is the total kinetic energy due to both his forward motion and his rotation?
d) What percentage of his kinetic energy is due to the rotation of his legs and arms?
A sphere of radius = 23.0 cm and mass m = 1.20 kg starts from rest and rolls without slipping down a 36.0 degree incline that is 13.0 m long.
a) Calculate its translational speed when it reaches the bottom.
b) Calculate its rotational speed when it reaches the bottom.
c) What is the ratio of translational to rotational kinetic energy at the bottom?
d) Does your answer in part A depend on mass or radius of the ball? Part B? Part C?
A bowling ball of mass 7.6 kg and radius 9.0 cm rolls without slipping down a lane at 3.6 m/s. Calculate its total kinetic energy.
Calculate the translational speed of a cylinder when it reaches the foot of an incline 7.30 m high. Assume it starts from rest and rolls without slipping.
A roller in a printing press turns through an angle θ(t) given by θ(t) = γt2 - βt3 , where γ = 3.20 rad/s2 and β = 0.500 rad/s3.
a) Calculate the angular velocity of the roller as a function of time.
b) Calculate the angular acceleration of the roller as a function of time.
c) What is the maximum positive angular velocity?
d) At what value of t does it occur?
A 2.30-m-long pole is balanced vertically on its tip. It starts to fall and its lower end does not slip. What will be the speed of the upper end of the pole just before it hits the ground? [Hint: Use conservation of energy.]
A merry-go-round has a mass of 1550 kg and a radius of 7.60 m. How much net work is required to accelerate it from rest to a rotation rate of 1.00 revolution per 9.00 s ? Assume it is a solid cylinder.
Find the moment of inertia of a hoop (a thin-walled, hollow ring) with mass and radius about an axis perpendicular to the hoops plane at an edge.
A hollow spherical shell has mass 8.20 kg and radius 0.225 m . It is initially at rest and then rotates about a stationary axis that lies along a diameter with a constant acceleration of 0.895 rad/s2 . What is the kinetic energy of the shell after it has turned through 6.25 rev ?
A fan blade rotates with angular velocity given by ωz(t) = γ - βt2.
a) Calculate the angular acceleration as a function of time.
b) If γ = 5.05 rad/s and β = 0.805 rad/s3 , calculate the instantaneous angular acceleration αz at t = 3.10 s .
c) If γ = 5.05 rad/s and β = 0.805 rad/s3 , calculate the average angular acceleration αav - z for the time interval t = 0 to t = 3.10 s .
Small blocks, each with mass m , are clamped at the ends and at the center of a rod of length L and negligible mass. Compute the moment of inertia of the system about an axis perpendicular to the rod and passing through:
a) the center of the rod
b) a point one-fourth of the length from one end
An airplane propeller is rotating at 1910 rev/min.
a) Compute the propellers angular velocity in rad/s.
b) How long in seconds does it take for the propeller to turn through 36°?
At t=0 a grinding wheel has an angular velocity of 26.0 rad/s . It has a constant angular acceleration of 31.0 rad/s2 until a circuit breaker trips at time t = 2.10 s. From then on, it turns through an angle 436 rad as it coasts to a stop at constant angular acceleration.
a) Through what total angle did the wheel turn between t=0 and the time it stopped?
b) At what time did it stop?
c) What was its acceleration as it slowed down?
The angular acceleration of a wheel, as a function of time, is α = 5.0 t2 - 8.5 t, where α is in rad/s2 and t is in seconds. If the wheel starts from rest (θ = 0, ω = 0, at t = 0), determine a formula for
a) the angular velocity ω as a function of time
b) the angular position θ as a function of time
c) evaluate ω at t = 4.0 s
d) evaluate θ at t = 4.0 s
A uniform bar has two small balls glued to its ends. The bar is 2.00 m long and has mass 6.00 kg , while the balls each have mass 0.300 kg and can be treated as point masses. Find the moment of inertia of this combination about an axis:
a) perpendicular to the bar through its center
b) perpendicular to the bar through one of the balls
c) parallel to the bar through both balls
d) parallel to the bar and 0.500 m from it
a) Calculate the angular velocity of the second hand of a clock. State in rad/s.
b) Calculate the angular velocity of the minute hand of a clock. State in rad/s.
c) Calculate the angular velocity of the hour hand of a clock. State in rad/s.
d) What is the angular acceleration in each case?