Angular momentum and rotational kinetic energy

 

1. (a) State the law of conservation of angular momentum.

(b)   State the expression for the angular momentum of an object in terms of its moment of inertia.

(c)   State the equation for the rotational kinetic energy of a rigid object.

 

2. A bicycle wheel has a moment of inertia of 0.25 kgm² about its hub.

Calculate the angular momentum of the wheel when rotating at 120 r.p.m.

 

3. A model aeroplane is flying in a horizontal circle at the end of a light wire. The mass of the aeroplane is 2.0 kg.  The radius of the circular path is 20 m.  The aeroplane makes 40 revolutions in one minute.

(a)   Calculate the linear velocity of the aeroplane.

(b)   Find the angular momentum of the aeroplane about the centre of the circle.

(c)   The wire suddenly breaks.  What is the new angular momentum of the aeroplane about the centre of the circle?

 

4. A shaft has a moment of inertia of 20 kgm² about its central axis. The shaft is rotating at 10 rpm.  This shaft is locked onto another shaft, which is initially stationary.  The second shaft has a moment of inertia of 30 kgm².

(a)   Find the angular momentum of the combination after the shafts are locked together.

(b)   What is the angular velocity of the combination after the shafts are locked together?

 

5. Which of the following are vector quantities:

torque, moment of inertia, angular velocity, tangential force, angular acceleration, rotational kinetic energy, radius of a circular motion.

 

6. Two children are playing on a roundabout of mass 250 kg. The roundabout can be considered to be a solid disc of diameter 3.0 m.  (I_disc = ½ MR²)

One child of mass 40 kg stands on the rim of the roundabout. The other child of mass 60 kg is positioned half way between the rim and the centre.

(a) Calculate the total moment of inertia of the roundabout and children.

(b) Determine the rotational kinetic energy of this system when it is rotating at 35 rpm.

 

7. A disc has a moment of inertia of 2.5 kgm². The disc is rotating at 2.0 rads^-1.

(a) Calculate the kinetic energy of the disc.

(b) How much energy needs to be supplied to increase its angular velocity to 15 rads^-1?

 

8. A solid cylinder and a hollow cylinder have the same mass and the same radius.

(a)   Which one has the larger moment of inertia about the central axis as shown opposite? You must justify your answer. (b)   The cylinders do not have the same length.  Does this affect your answer to part (a)?  Again you must justify your answer.

 

9. A cylinder of mass 3.0 kg rolls down a slope without slipping. The radius R of the cylinder is 50 mm and its moment of inertia is ½MR².  The slope has a length of 0.30 m and is inclined at 40^o to the horizontal.

(a)   Calculate the loss in gravitational potential energy as the cylinder rolls from the top of the slope to the bottom of the slope.

(b)   Find the speed with which the cylinder reaches the bottom of the slope.

 

10. A turntable is rotating freely at 40 rpm about a vertical axis. A small mass of 50 g falls vertically onto the turntable and lands at a distance of 80 mm from the central axis.  The rotation of the turntable is reduced to 33 rpm. Find the moment of inertia of the turntable.

 

11. An LP of mass 180g and diameter 30 cm is dropped onto a rotating turntable. The turntable has a moment of inertia about its axis of rotation of 5.0 x 10^-3 kgm².  The turntable was initially rotating at 3.5 rads^-1. Determine the common angular velocity of the turntable and the LP.

 

12. A skater with her arms pulled in has a moment of inertia of 1.5 kgm² about a vertical axis through the centre of her body. With her arms outstretched the moment of inertia is increased to 10 kgm². With her arms pulled in, the skater is spinning at 30 rads^-1.  The skater then extends her arms.

(a)   Calculate her final angular speed.

(b)   Find the change in kinetic energy.

(c)   Explain why there is a change in kinetic energy.

 

13. A skater is spinning at 3.0 rads^-1 with her arms and one leg outstretched.

The angular speed is increased to 25 rads^-1 when she draws her arms and leg in.

(a)   Explain why this movement of her arms and leg affects the rotational speed.

(b)   Her moment of inertia about her spin axis is 5.0 kgm² with her arms and leg outstretched.  Calculate her moment of inertia when her arms and leg are drawn in.

 

14. A roundabout has a moment of inertia of 300 kgm² about its axis of rotation. Three children, each of mass 20 kg, stand 2.0 m from the centre of the stationary roundabout.  They all start to run round the roundabout in the same direction until they reach a speed of 3.0 ms^-1 relative to the roundabout. Calculate the angular velocity of the roundabout.

 

15. A disc is rotating at 100 rpm in a horizontal plane about a vertical axis. A small piece of plasticene is dropped vertically onto the disc and sticks at a position 50 mm from the centre of the disc.  The plasticene has a mass of 20 g.  The disc is slowed to 75 rpm.  Calculate the moment of inertia of the disc.

 

16. The radius of a spherical neutron star is 20 km. The star rotates at 1800 rpm.

(a)   Calculate the velocity of a point on the equator of the star.

(b)   The mass of the neutron star is the same as the mass of the sun. What is the density of the neutron star?

(c)   The radius of a neutron is about 10^-15 m.  Estimate the average spacing of the neutrons in the neutron star.