6.05b Circular motion: v=r*omega and a=v^2/r

3 A particle \(P\) of mass 3.5 kg is attached to one end of a light elastic string of natural length 0.8 m and modulus of elasticity 75 N . The other end of the string is attached to a fixed point \(O\). The particle rotates in a horizontal circle with a constant angular velocity of \(3 \mathrm { rad } \mathrm { s } ^ { - 1 }\). The centre of the circle is vertically below \(O\). The magnitude of the tension in the string is \(T \mathrm {~N}\) and the length of the extended string is \(L \mathrm {~m}\) (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{a8c9d007-e67f-4637-9e74-630ba9a91442-3_460_424_447_817}

1. By considering the acceleration of \(P\), show that \(T = 31.5 L\).
2. Write down another relationship between \(T\) and \(L\).
3. Find the value of \(T\) and the value of \(L\).
4. Find the angle that the string makes with the downwards vertical through \(O\).

7 A smooth track \(A B\) is in the shape of an arc of a circle with centre \(O\) and radius 1.4 m . The track is fixed in a vertical plane with \(A\) above the level of \(B\) and a point \(C\) on the track vertically below \(O\). Angle \(A O C\) is \(60 ^ { \circ }\) and angle \(C O B\) is \(30 ^ { \circ }\). Point \(C\) is 2.5 m vertically above the point \(F\), which lies in a horizontal plane. A particle of mass 0.4 kg is placed at \(A\) and projected down the track with an initial velocity of \(0.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The particle first hits the plane at point \(H\) (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{a8c9d007-e67f-4637-9e74-630ba9a91442-5_767_1265_488_415}

1. Find the magnitude of the contact force between the particle and the track when the particle is at \(B\). [5]
2. Find the distance \(F H\).

1 A particle \(P\) of mass 2.4 kg is attached to one end of a light inextensible string of length 1.4 m . The other end of the string is attached to a fixed point \(O\) on a smooth horizontal table. \(P\) moves on the table at constant speed along a circular path with \(O\) at its centre. The magnitude of the tension in the string is 21 N .

1. (a) Find the magnitude of the acceleration of \(P\).
   (b) State the direction of the acceleration of \(P\).
2. Find the speed of \(P\).
3. Find the time taken for \(P\) to complete a single revolution.

5 A particle, of mass 6 kg , is attached to one end of a light inextensible string. The other end of the string is attached to the fixed point \(O\). The particle is set in motion, so that it moves in a horizontal circle at constant speed, with the string at an angle of \(30 ^ { \circ }\) to the vertical. The centre of this circle is vertically below \(O\). \includegraphics[max width=\textwidth, alt={}, center]{851cb2a3-5bc8-4af9-b1fc-a143d37beebe-4_586_490_541_767} The particle moves in a horizontal circle with an angular speed of 40 revolutions per minute.

1. Show that the angular speed of the particle is \(\frac { 4 \pi } { 3 }\) radians per second.
2. Show that the tension in the string is 67.9 N , correct to three significant figures.
3. Find the radius of the horizontal circle.

6 A particle, of mass 4 kg , is attached to one end of a light inextensible string of length 1.2 metres. The other end of the string is attached to a fixed point \(O\). The particle moves in a horizontal circle at a constant speed. The angle between the string and the vertical is \(\theta\). \includegraphics[max width=\textwidth, alt={}, center]{06b431ca-d3a8-46d6-b9f8-bac08d3fd51e-4_529_554_1580_737}

1. Find the radius of the horizontal circle in terms of \(\theta\).
2. The angular speed of the particle is 5 radians per second. Find \(\theta\).

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.

4 Two light inextensible strings each have one end attached to a particle, \(P\), of mass 6 kg . The other ends of the strings are attached to the fixed points \(B\) and \(C\). The point \(C\) is vertically above the point \(B\). The particle moves, at constant speed, in a horizontal circle, with centre 0.6 m below point \(B\), with the strings inclined at \(40 ^ { \circ }\) and \(60 ^ { \circ }\) to the vertical, as shown in the diagram. Both strings are taut. \includegraphics[max width=\textwidth, alt={}, center]{9cfa110c-ee11-447a-b21a-3f436432e27d-4_761_542_539_751}

1. As the particle moves in the horizontal circle, the tensions in the two strings are equal. Show that the tension in the strings is 46.4 N , correct to three significant figures.
2. Find the speed of the particle.

5 J
10 J
20 J 4 Reena is skating on an ice rink, which has a horizontal surface. She follows a circular path of radius 5 metres and centre \(O\) She completes 10 full revolutions in 1 minute, moving with a constant angular speed of \(\omega\) radians per second. The mass of Reena is 40 kg
4

1. Find the value of \(\omega\) 4
2. (i) Find the magnitude of the horizontal resultant force acting on Reena.
   4 (b) (ii) Show the direction of this horizontal resultant force on the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{78120346-4a16-4545-925a-d6fab4b750e9-03_380_442_2017_861} 5 An impulse of \(\left[ \begin{array} { r } - 5 \\ 12 \end{array} \right] \mathrm { N } \mathrm { s }\) is applied to a particle of mass 5 kg which is moving with velocity \(\left[ \begin{array} { l } 6 \\ 2 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\) 5 (a) Calculate the magnitude of the impulse. 5 (b) Find the speed of the particle immediately after the impulse is applied.

20 J 4 Reena is skating on an ice rink, which has a horizontal surface. She follows a circular path of radius 5 metres and centre \(O\) She completes 10 full revolutions in 1 minute, moving with a constant angular speed of \(\omega\) radians per second. The mass of Reena is 40 kg
4

1. Find the value of \(\omega\) 4
2. (i) Find the magnitude of the horizontal resultant force acting on Reena.
   4 (b) (ii) Show the direction of this horizontal resultant force on the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{78120346-4a16-4545-925a-d6fab4b750e9-03_380_442_2017_861} 5 An impulse of \(\left[ \begin{array} { r } - 5 \\ 12 \end{array} \right] \mathrm { N } \mathrm { s }\) is applied to a particle of mass 5 kg which is moving with velocity \(\left[ \begin{array} { l } 6 \\ 2 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\) 5 (a) Calculate the magnitude of the impulse. 5 (b) Find the speed of the particle immediately after the impulse is applied.
   6 A ball is thrown with speed \(u\) at an angle of \(45 ^ { \circ }\) to the horizontal from a point \(O\) When the horizontal displacement of the ball is \(x\), the vertical displacement of the ball above \(O\) is \(y\) where $$y = x - \frac { k x ^ { 2 } } { u ^ { 2 } }$$ 6 (a) Use dimensional analysis to find the dimensions of \(k\) 6 (b) State what can be deduced about \(k\) from the dimensions that you found in part (a).
   7 Two smooth, equally sized spheres, \(A\) and \(B\), are moving in the same direction along a straight line on a smooth horizontal surface, as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{78120346-4a16-4545-925a-d6fab4b750e9-06_314_465_420_849} The spheres subsequently collide.
   Immediately after the collision, \(A\) has speed \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(B\) has speed \(3.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) The coefficient of restitution between the spheres is \(e\) 7 (a) (i) Show that \(A\) does not change its direction of motion as a result of the collision.
   7 (a) (ii) Find the value of \(e\) 7 (b) Given that the mass of \(B\) is 0.6 kg , find the mass of \(A\)

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.
   6
2. Prove that the magnitude of the resultant force on the particle is constant.

4 A car of mass 1400 kg drives around a horizontal circular bend of radius 60 metres.
The car has a constant speed of \(12 \mathrm {~ms} ^ { - 1 }\) on the bend.
Calculate the magnitude of the resultant force acting on the car.
[0pt] [2 marks] \(5 \quad\) A region bounded by the curve with equation \(y = 4 - x ^ { 2 }\), the \(x\)-axis and the \(y\)-axis is shown below. \includegraphics[max width=\textwidth, alt={}, center]{cd0d239b-ab92-4d17-9cb8-45722e2894cb-04_641_380_408_831} The region is rotated through \(360 ^ { \circ }\) around the \(x\)-axis to create a uniform solid.

3
Two particles \(A\) and \(B\) are connected by a light inextensible string. Particle \(A\) has mass 1.2 kg and moves on a smooth horizontal table in a circular path of radius 0.6 m and centre \(O\). The string passes through a small smooth hole at \(O\). Particle \(B\) moves in a horizontal circle in such a way that it is always vertically below \(A\). The angle that the portion of the string below the table makes with the downwards vertical through \(O\) is \(\theta\), where \(\cos \theta = \frac { 4 } { 5 }\) (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{75f629e7-969d-43ae-8222-031875ae54ae-02_453_696_1571_552}

1. Find the time taken for the particles to perform a complete revolution.
2. Find the mass of \(B\).

9 \includegraphics[max width=\textwidth, alt={}, center]{d8ca5464-435f-45e0-8e19-1830415a7c60-4_666_816_1384_662} A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string of length \(l\). The other end of the string is attached to a fixed point \(A\). The particle moves with constant angular speed \(\omega\) in a horizontal circle whose centre is at a distance \(h\) vertically below \(A\) (see diagram).

1. Show that however fast the particle travels \(A P\) will never become horizontal, and that the tension in the string is always greater than the weight of the particle.
2. Find the tension in the string in terms of \(m , l\) and \(\omega\).
3. Show that \(\omega ^ { 2 } h = g\) and calculate \(\omega\) when \(h\) is 0.5 m .

9 \includegraphics[max width=\textwidth, alt={}, center]{a19fab61-da1c-4803-9dbc-38d618a0c58e-5_671_817_255_623} A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string of length \(l\). The other end of the string is attached to a fixed point \(A\). The particle moves with constant angular speed \(\omega\) in a horizontal circle whose centre is at a distance \(h\) vertically below \(A\) (see diagram).

1. Show that however fast the particle travels \(A P\) will never become horizontal, and that the tension in the string is always greater than the weight of the particle.
2. Find the tension in the string in terms of \(m , l\) and \(\omega\).
3. Show that \(\omega ^ { 2 } h = g\) and calculate \(\omega\) when \(h\) is 0.5 m .

8 \includegraphics[max width=\textwidth, alt={}, center]{c4bbba86-2968-4247-b300-357217cf213b-4_670_819_548_621} A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string of length \(l\). The other end of the string is attached to a fixed point \(A\). The particle moves with constant angular speed \(\omega\) in a horizontal circle whose centre is at a distance \(h\) vertically below \(A\) (see diagram).

1. Find the tension in the string in terms of \(m , l\) and \(\omega\).
2. Show that \(\omega ^ { 2 } h = g\).
3. Deduce an expression in terms of \(g\) and \(h\) for the time taken for \(P\) to complete one full circle during its motion.

8 \includegraphics[max width=\textwidth, alt={}, center]{adf5bd3c-5408-421d-b7d5-dea2d0f0185b-4_604_734_1512_667} A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string of length \(l\). The other end of the string is attached to a fixed point \(A\). The particle moves with constant angular speed \(\omega\) in a horizontal circle whose centre is at a distance \(h\) vertically below \(A\) (see diagram).

1. Find the tension in the string in terms of \(m , l\) and \(\omega\).
2. Show that \(\omega ^ { 2 } h = g\).
3. Deduce an expression in terms of \(g\) and \(h\) for the time taken for \(P\) to complete one full circle during its motion.

8 \includegraphics[max width=\textwidth, alt={}, center]{f4acd242-eb78-4124-bfa2-fdecaa188690-4_614_741_1548_662} A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string of length \(l\). The other end of the string is attached to a fixed point \(A\). The particle moves with constant angular speed \(\omega\) in a horizontal circle whose centre is at a distance \(h\) vertically below \(A\) (see diagram).

1. Find the tension in the string in terms of \(m , l\) and \(\omega\).
2. Show that \(\omega ^ { 2 } h = g\).
3. Deduce an expression in terms of \(g\) and \(h\) for the time taken for \(P\) to complete one full circle during its motion.

\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 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]

\includegraphics{figure_4} Three identical uniform discs, \(A\), \(B\) and \(C\), each have mass \(m\) and radius \(a\). They are joined together by uniform rods, each of which has mass \(\frac{1}{4}m\) and length \(2a\). The discs lie in the same plane and their centres form the vertices of an equilateral triangle of side \(4a\). Each rod has one end rigidly attached to the circumference of a disc and the other end rigidly attached to the circumference of an adjacent disc, so that the rod lies along the line joining the centres of the two discs (see diagram).

1. Find the moment of inertia of this object about an axis \(l\), which is perpendicular to the plane of the object and through the centre of disc \(A\). [6]

The object is free to rotate about the horizontal axis \(l\). It is released from rest in the position shown, with the centre of disc \(B\) vertically above the centre of disc \(A\).

1. Write down the change in the vertical position of the centre of mass of the object when the centre of disc \(B\) is vertically below the centre of disc \(A\). Hence find the angular velocity of the object when the centre of disc \(B\) is vertically below the centre of disc \(A\). [4]

\includegraphics{figure_4} Three identical uniform discs, \(A\), \(B\) and \(C\), each have mass \(m\) and radius \(a\). They are joined together by uniform rods, each of which has mass \(\frac{1}{3}m\) and length \(2a\). The discs lie in the same plane and their centres form the vertices of an equilateral triangle of side \(4a\). Each rod has one end rigidly attached to the circumference of a disc and the other end rigidly attached to the circumference of an adjacent disc, so that the rod lies along the line joining the centres of the two discs (see diagram).

1. Find the moment of inertia of this object about an axis \(l\), which is perpendicular to the plane of the object and through the centre of disc \(A\). [6]
2. The object is free to rotate about the horizontal axis \(l\). It is released from rest in the position shown, with the centre of disc \(B\) vertically above the centre of disc \(A\). Write down the change in the vertical position of the centre of mass of the object when the centre of disc \(B\) is vertically below the centre of disc \(A\). Hence find the angular velocity of the object when the centre of disc \(B\) is vertically below the centre of disc \(A\). [4]

\includegraphics{figure_1} A uniform disc with centre \(O\), mass \(m\) and radius \(a\) is free to rotate without resistance in a vertical plane about a horizontal axis through \(O\). One end of a light inextensible string is attached to the rim of the disc and wrapped around the rim. The other end of the string is attached to a block of mass \(3m\) (see diagram). The system is released from rest with the block hanging vertically. While the block is in motion, it experiences a constant vertical resisting force of magnitude \(0.9mg\). Find the tension in the string in terms of \(m\) and \(g\). [5]

A particle \(P\) is moving in a circle of radius 2 m. At time \(t\) seconds, its velocity is \((t - 1)^2\) m s\(^{-1}\). At a particular time \(T\) seconds, where \(T > 0\), the magnitude of the radial component of the acceleration of \(P\) is 8 m s\(^{-2}\). Find the magnitude of the transverse component of the acceleration of \(P\) at this instant. [5]