Position vector circular motion

1 A line \(O P\) of fixed length \(l\) rotates in a plane about the fixed point \(O\). At time \(t = 0\), the line is at the position \(O A\). At time \(t\), angle \(A O P = \theta\) radians and \(\frac { \mathrm { d } \theta } { \mathrm { d } t } = \sin \theta\). Show that, for all \(t\), the magnitude of the acceleration of \(P\) is equal to the magnitude of its velocity.

4 A particle moves on a horizontal plane, in which the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular. At time \(t\), the particle's position vector, \(\mathbf { r }\), is given by $$\mathbf { r } = 4 \cos 3 t \mathbf { i } - 4 \sin 3 t \mathbf { j }$$

1. Prove that the particle is moving on a circle, which has its centre at the origin.
2. Find an expression for the velocity of the particle at time \(t\).
3. Find an expression for the acceleration of the particle at time \(t\).
4. The acceleration of the particle can be written as $$\mathbf { a } = k \mathbf { r }$$ where \(k\) is a constant. Find the value of \(k\).
5. State the direction of the acceleration of the particle.

5 Tom is on a fairground ride.
Tom's position vector, \(\mathbf { r }\) metres, at time \(t\) seconds is given by $$\mathbf { r } = 2 \cos t \mathbf { i } + 2 \sin t \mathbf { j } + ( 10 - 0.4 t ) \mathbf { k }$$ The perpendicular unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are in the horizontal plane and the unit vector \(\mathbf { k }\) is directed vertically upwards.

1. 1. Find Tom's position vector when \(t = 0\).
   2. Find Tom's position vector when \(t = 2 \pi\).
   3. Write down the first two values of \(t\) for which Tom is directly below his starting point.
2. Find an expression for Tom's velocity at time \(t\).
3. Tom has mass 25 kg . Show that the resultant force acting on Tom during the motion has constant magnitude. State the magnitude of the resultant force.
   (5 marks)

5 A particle moves on a horizontal plane in which the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are directed east and north respectively. At time \(t\) seconds, the particle's position vector, \(\mathbf { r }\) metres, is given by $$\mathbf { r } = 8 \left( \cos \frac { 1 } { 4 } t \right) \mathbf { i } - 8 \left( \sin \frac { 1 } { 4 } t \right) \mathbf { j }$$

1. Find an expression for the velocity of the particle at time \(t\).
2. Show that the speed of the particle is a constant.
3. Prove that the particle is moving in a circle.
4. Find the angular speed of the particle.
5. Find an expression for the acceleration of the particle at time \(t\).
6. State the magnitude of the acceleration of the particle.

6 A particle moves with constant speed on a circular path of radius 2 metres. The centre of the circle has position vector \(2 \mathbf { j }\) metres.
At time \(t = 0\), the particle is at the origin and is moving in the positive \(\mathbf { i }\) direction.
The particle returns to the origin every 4 seconds.
The unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular.
6

1. Calculate the angular speed of the particle.
   6
2. Write down an expression for the position vector of the particle at time \(t\) seconds.
   6
3. Find an expression for the acceleration of the particle at time \(t\) seconds.
   6
4. State the magnitude of the acceleration of the particle.
   6
5. State the time when the acceleration is first directed towards the origin.

6 A particle, of mass 5 kg , moves on a circular path so that at time \(t\) seconds it has position vector \(\mathbf { r }\) metres, where $$\mathbf { r } = ( 2 \sin 3 t ) \mathbf { i } + ( 2 \cos 3 t ) \mathbf { j }$$ 6

1. Prove that the velocity of the particle is perpendicular to its position vector.
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2. Prove that the magnitude of the resultant force on the particle is constant.