Radial and transverse acceleration

4
Two coplanar discs, of radii 0.5 m and 0.3 m , rotate about their centres \(A\) and \(B\) respectively, where \(A B = 0.8 \mathrm {~m}\). At time \(t\) seconds the angular speed of the larger disc is \(\frac { 1 } { 2 } t \mathrm { rad } \mathrm { s } ^ { - 1 }\) (see diagram). There is no slipping at the point of contact. For the instant when \(t = 2\), find

1. the angular speed of the smaller disc,
2. the magnitude of the acceleration of a point \(P\) on the circumference of the larger disc, and the angle between the direction of this acceleration and \(P A\).

1 A particle \(P\) is moving in a circle of radius 0.25 m . At time \(t\) seconds, its velocity is \(\left( 2 t ^ { 2 } - 4 t + 3 \right) \mathrm { m } \mathrm { s } ^ { - 1 }\). At a particular time \(T\) seconds, where \(T > 0\), the magnitude of the transverse component of the acceleration of \(P\) is \(6 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Find the magnitude of the radial component of the acceleration of \(P\) at this instant.

1 A particle \(P\) is moving in a fixed circle of radius 0.8 m . At time \(t \mathrm {~s}\) its velocity is \(\left( t ^ { 2 } - t + 2 \right) \mathrm { m } \mathrm { s } ^ { - 1 }\). Find the magnitudes of the radial and the transverse components of the acceleration of \(P\) when \(t = 2\). Radial component
Transverse component \(\_\_\_\_\)

1 A particle \(P\) moves along an arc of a circle with centre \(O\) and radius 2 m . At time \(t\) seconds, the angle POA is \(\theta\), where \(\theta = 1 - \cos 2 t\), and \(A\) is a fixed point on the arc of the circle.

1. Show that the magnitude of the radial component of the acceleration of \(P\) when \(t = \frac { 1 } { 6 } \pi\) is \(6 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). \includegraphics[max width=\textwidth, alt={}, center]{2aaf3493-6509-4668-91a2-9f4708bbbb58-03_65_1573_488_324}
2. Find the magnitude of the transverse component of the acceleration of \(P\) when \(t = \frac { 1 } { 6 } \pi\).

1 A particle is moving in a circle of radius 2 m . At time \(t \mathrm {~s}\) its velocity is \(\left( t ^ { 2 } - 12 \right) \mathrm { m } \mathrm { s } ^ { - 1 }\). Find the magnitude of the resultant acceleration of the particle when \(t = 4\).

1 A particle \(P\) is moving in a circle of radius 0.8 m . At time \(t \mathrm {~s}\) its velocity is \(\left( 8 - p t + t ^ { 2 } \right) \mathrm { m } \mathrm { s } ^ { - 1 }\), where \(p\) is a constant. The magnitude of the transverse component of the acceleration of \(P\) when \(t = 2\) is zero. Find the magnitude of the radial component of the acceleration of \(P\) when \(t = 2\).

1 A small smooth ring of mass 0.5 kg is travelling round a smooth circular wire, with centre O and radius 0.8 m . The circle of wire is in a horizontal plane. The speed of the ring, \(v \mathrm {~ms} ^ { - 1 }\), at time \(t \mathrm {~s}\) after passing through a point A on the wire is given by \(\mathrm { v } = 0.2 \mathrm { t } ^ { 2 } + 0.4 \mathrm { t } + 0.1\).

1. Find the angular speed of the ring 5 seconds after it passes through A .
2. Find the distance the ring travels along the wire in the first second after passing through A . At time \(T\) s after the ring passes through A the magnitude of the force exerted on the ring by the wire is 6.4 N . You may assume that any forces acting on the ring other than the force exerted on the ring by the wire and gravity can be ignored.
3. 1. Determine the value of \(T\).
   2. Hence find the tangential acceleration of the ring at this time.

8 A horizontal turntable rotates about a vertical axis. Starting from rest, it accelerates uniformly to an angular velocity of \(8.4 \mathrm { rad } \mathrm { s } ^ { - 1 }\) in 2 s .

1. Find the angular acceleration of the turntable.
2. A particle rests on the turntable at a distance of 0.15 m from the axis. Find the radial and transverse components of the acceleration of the particle when the angular velocity is \(1.2 \mathrm { rad } \mathrm { s } ^ { - 1 }\). Find also the magnitude of the acceleration at this instant.

\includegraphics{figure_4} Two coplanar discs, of radii \(0.5\) m and \(0.3\) m, rotate about their centres \(A\) and \(B\) respectively, where \(AB = 0.8\) m. At time \(t\) seconds the angular speed of the larger disc is \(\frac{1}{2}t\) rad s\(^{-1}\) (see diagram). There is no slipping at the point of contact. For the instant when \(t = 2\), find

1. the angular speed of the smaller disc, [2]
2. the magnitude of the acceleration of a point \(P\) on the circumference of the larger disc, and the angle between the direction of this acceleration and \(PA\). [7]

A particle \(P\) moves along an arc of a circle with centre \(O\) and radius 2 m. At time \(t\) seconds, the angle \(POA\) is \(\theta\), where \(\theta = 1 - \cos 2t\), and \(A\) is a fixed point on the arc of the circle.

1. Show that the magnitude of the radial component of the acceleration of \(P\) when \(t = \frac{1}{6}\pi\) is 6 m s\(^{-2}\). [2]
2. Find the magnitude of the transverse component of the acceleration of \(P\) when \(t = \frac{1}{6}\pi\). [2]

A roundabout in a playground can be modeled as a horizontal circular platform with centre \(O\). The roundabout is free to rotate about a vertical axis through \(O\). A child sits without slipping on the roundabout at a horizontal distance of 1.5 m from \(O\) and completes one revolution in 2.4 seconds.

1. Calculate the speed of the child. [3]
2. Find the magnitude and direction of the acceleration of the child. [3]