Determine the final angular speed of the probe–satellite system.

Explain why the angular speed will increase.

Determine the angular velocity of the wheel at B.

Describe the effect of F on the angular speed of the wheel.

After time t the rod makes an angle θ with the horizontal. Outline why the equation $\theta =\frac{1}{2}\alpha t^2$ cannot be used to find the time it takes θ to become $\frac{\pi }{2}$ (that is for the rod to become vertical for the first time).

In moving from point A to point B, the centre of mass of the wheel falls through a vertical distance of 0.36 m. Show that the translational speed of the wheel is about 1 m s^–1 after its displacement.

Calculate, for the merry-go-round after one revolution, the angular momentum.

Deduce the linear acceleration of the centre of mass of the probe.

The velocity of the falling object is 1.89 m s^–1 at 3.98 s. Calculate the average angular acceleration of the flywheel.

At the instant the rod becomes vertical show that the angular speed is ω = 2.43 rad s^–1.

Calculate, in rad s^–2, the initial angular acceleration $\alpha$ of the rod.

Calculate the maximum tension in the string.

State the relationship between the force of friction and the angle of the incline.

Show that the torque acting on the flywheel is about 0.3 Nm.

Calculate the number of revolutions made by the system before it comes to rest.

The moment of inertia of the wheel is 1.3 × 10^–4 kg m². Outline what is meant by the moment of inertia.

Calculate the new angular speed of the rotating system.

Calculate F.

Calculate, for the merry-go-round after one revolution, the angular speed. 

Show that the final angular velocity of the bar is about $3 \mathrm{rad} {\mathrm{s}}^{-1}$.

(i) Calculate the tension in the string.

(ii) Determine the mass m of the falling object.

The angle of the incline is slowly increased from zero. Determine the angle, in terms of the coefficient of friction, at which the hoop will begin to slip.

At t = 5.00 s the flywheel is spinning with angular velocity 200 rad s^–1. The support bearings exert a constant frictional torque on the axle. The flywheel comes to rest after 8.00 × 10³ revolutions. Calculate the magnitude of the frictional torque exerted on the flywheel.

Show that the total kinetic energy E_k of the sphere when it rolls, without slipping, at speed v is $E_{\text{K}}=\frac{7}{10}m{v}^2$.

The torque is removed. The bar comes to rest in $30$ complete rotations with constant angular deceleration. Determine the time taken for the bar to come to rest.

Draw the variation with time $t$ of the angular displacement $\theta$ of the bar during the acceleration.

Explain the direction in which the person-turntable system starts to rotate.

Calculate the torque acting on the bar while it is accelerating.

A solid sphere of radius $r$ and mass $m$ is released from rest and rolls down a slope, without slipping. The vertical height of the slope is $h$. The moment of inertia $I$ of this sphere about an axis through its centre is $\frac{2}{5}m{r}^2$.

Show that the linear velocity $v$ of the sphere as it leaves the slope is $\sqrt{\frac{10gh}{7}}$.

The forces act for 2.00 s. Show that the final angular speed of the probe is about 16 rad$$s^–1.

Show that the linear acceleration a of the hoop is given by the equation shown.

a = $\frac{g\times \sin q}{2}$

Calculate the work done by the child in moving from the edge to the centre.

Write down an expression, in terms of M, v and R, for the angular momentum of the system about the vertical axis just before the collision.

Just after the collision the system begins to rotate about the vertical axis with angular velocity ω. Show that the angular momentum of the system is equal to $\frac{4}{3}M{R}^2\omega$.

Hence, show that $\omega =\frac{v}{4R}$.

Show that the angular deceleration of the system is 0.043 rad$$s^–2.

Calculate the resultant torque about the axis of the probe.

Determine in terms of M and v the energy lost during the collision.

Calculate the loss of rotational kinetic energy due to the linking of the probe with the satellite.

Show that the angular acceleration of the merry-go-round is 0.2 rad s^–2.

A solid sphere of mass 1.5 kg is rolling, without slipping, on a horizontal surface with a speed of 0.50 m s^-1. The sphere then rolls, without slipping, down a ramp to reach a horizontal surface that is 45 cm lower.

Calculate the speed of the sphere at the bottom of the ramp.