Angular kinematics – constant angular acceleration/deceleration

1 A wheel rotating about a fixed axis is slowing down with constant angular deceleration. Initially the angular speed is \(24 \mathrm { rad } \mathrm { s } ^ { - 1 }\). In the first 5 seconds the wheel turns through 96 radians.

1. Find the angular deceleration.
2. Find the total angle the wheel turns through before coming to rest.

1 A wheel is rotating about a fixed axis, and is slowing down with constant angular deceleration \(0.3 \mathrm { rad } \mathrm { s } ^ { - 2 }\).

1. Find the angle the wheel turns through as its angular speed changes from \(8 \mathrm { rad } \mathrm { s } ^ { - 1 }\) to \(5 \mathrm { rad } \mathrm { s } ^ { - 1 }\).
2. Find the time taken for the wheel to make its final complete revolution before coming to rest.

1 A propeller shaft has constant angular acceleration. It turns through 160 radians as its angular speed increases from \(15 \mathrm { rad } \mathrm { s } ^ { - 1 }\) to \(25 \mathrm { rad } \mathrm { s } ^ { - 1 }\). Find

1. the angular acceleration of the propeller shaft,
2. the time taken for this increase in angular speed.

1 The driveshaft of an electric motor begins to rotate from rest and has constant angular acceleration. In the first 8 seconds it turns through 56 radians.

1. Find the angular acceleration.
2. Find the angle through which the driveshaft turns while its angular speed increases from \(20 \mathrm { rad } \mathrm { s } ^ { - 1 }\) to \(36 \mathrm { rad } \mathrm { s } ^ { - 1 }\).

2 A rotating turntable is slowing down with constant angular deceleration. It makes 16 revolutions as its angular speed decreases from \(8 \mathrm { rad } \mathrm { s } ^ { - 1 }\) to rest.

1. Find the angular deceleration of the turntable.
2. Find the angular speed of the turntable at the start of its last complete revolution before coming to rest.
3. Find the time taken for the turntable to make its last complete revolution before coming to rest.

1 A top is set spinning with initial angular speed \(83 \mathrm { rad } \mathrm { s } ^ { - 1 }\), and it slows down with constant angular deceleration. When it has turned through 1000 radians, its angular speed is \(67 \mathrm { rad } \mathrm { s } ^ { - 1 }\).

1. Find the angular deceleration of the top.
2. Find the time taken, from the start, for the top to turn through 400 radians.

1 A wheel is rotating and is slowing down with constant angular deceleration. The initial angular speed is \(80 \mathrm { rad } \mathrm { s } ^ { - 1 }\), and after 15 s the wheel has turned through 1020 radians.

1. Find the angular deceleration of the wheel.
2. Find the angle through which the wheel turns in the last 5 s before it comes to rest.
3. Find the total number of revolutions made by the wheel from the start until it comes to rest.

1 When the power is turned off, a fan disk inside a jet engine slows down with constant angular deceleration \(0.8 \mathrm { rad } \mathrm { s } ^ { - 2 }\).

1. Find the time taken for the angular speed to decrease from \(950 \mathrm { rad } \mathrm { s } ^ { - 1 }\) to \(750 \mathrm { rad } \mathrm { s } ^ { - 1 }\).
2. Find the angle through which the disk turns as the angular speed decreases from \(220 \mathrm { rad } \mathrm { s } ^ { - 1 }\) to \(200 \mathrm { rad } \mathrm { s } ^ { - 1 }\).
3. Find the time taken for the disk to make the final 10 revolutions before coming to rest.

1 A uniform square lamina, of mass 4.5 kg and side 0.6 m , is rotating about a fixed vertical axis which is perpendicular to the lamina and passes through its centre. A stationary particle becomes attached to the lamina at one of its corners, and this causes the angular speed of the lamina to change instantaneously from \(2.2 \mathrm { rad } \mathrm { s } ^ { - 1 }\) to \(1.5 \mathrm { rad } \mathrm { s } ^ { - 1 }\).

1. Find the mass of the particle. The lamina then slows down with constant angular deceleration. It turns through 36 radians as its angular speed reduces from \(1.5 \mathrm { rad } \mathrm { s } ^ { - 1 }\) to zero.
2. Find the time taken for the lamina to come to rest.

1 A camshaft inside an engine is rotating with angular speed \(42 \mathrm { rads } ^ { - 1 }\). When the throttle is opened the camshaft speeds up with constant angular acceleration, and 8 seconds after the throttle was opened the angular speed is \(76 \mathrm { rad } \mathrm { s } ^ { - 1 }\).

1. Find the angular acceleration of the camshaft.
2. Find the time taken for the camshaft to turn through 810 radians from the moment that the throttle was opened.

1 A turntable is rotating at \(3 \mathrm { rad } \mathrm { s } ^ { - 1 }\). The turntable is then accelerated so that after 4 revolutions it is rotating at \(12.4 \mathrm { rad } \mathrm { s } ^ { - 1 }\). Assuming that the angular acceleration of the turntable is constant,

1. find the angular acceleration,
2. find the time taken to increase its angular speed from \(3 \mathrm { rad } \mathrm { s } ^ { - 1 }\) to \(12.4 \mathrm { rad } \mathrm { s } ^ { - 1 }\).

A circular flywheel of radius 0.3 m, with moment of inertia about its axis 18 kg m\(^2\), is rotating freely with angular speed 6 rad s\(^{-1}\). A tangential force of constant magnitude 48 N is applied to the rim of the flywheel, in order to slow the flywheel down. Find the time taken for the angular speed of the flywheel to be reduced to 2 rad s\(^{-1}\). [4]

A circular flywheel of radius \(0.3\) m, with moment of inertia about its axis \(18\) kg m\(^2\), is rotating freely with angular speed \(6\) rad s\(^{-1}\). A tangential force of constant magnitude \(48\) N is applied to the rim of the flywheel, in order to slow the flywheel down. Find the time taken for the angular speed of the flywheel to be reduced to \(2\) rad s\(^{-1}\). [4]

A circular wheel is modelled as a uniform disc of mass \(6\) kg and radius \(0.25\) m. It is rotating with angular speed \(2\) rad s\(^{-1}\) about a fixed smooth axis perpendicular to its plane and passing through its centre. A braking force of constant magnitude is applied tangentially to the rim of the wheel. The wheel comes to rest \(5\) s after the braking force is applied. Find the magnitude of the braking force and the angle turned through by the wheel while the braking force acts. [7]