6.05a Angular velocity: definitions

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 \(O\).

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 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.

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 \(A\)

2 A particle has an angular speed of 72 revolutions per minute.
Find the angular speed in radians per second.
Circle your answer.
[0pt] [1 mark] \(\frac { 6 \pi } { 5 } \quad \frac { 12 \pi } { 5 } \quad 12 \pi \quad 24 \pi\)

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.

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 .

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 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]

Answer only one of the following two alternatives. EITHER \includegraphics{figure_11a} The diagram shows a uniform thin rod \(AB\) of length \(3a\) and mass \(8m\). The end \(A\) is rigidly attached to the surface of a sphere with centre \(O\) and radius \(a\). The rod is perpendicular to the surface of the sphere. The sphere consists of two parts: an inner uniform solid sphere of mass \(m\) and radius \(a\) surrounded by a thin uniform spherical shell of mass \(m\) and also of radius \(a\). The horizontal axis \(l\) is perpendicular to the rod and passes through the point \(C\) on the rod where \(AC = a\).

1. Show that the moment of inertia of the object, consisting of rod, shell and inner sphere, about the axis \(l\) is \(\frac{289}{15}ma^2\). [6]

The object is free to rotate about the axis \(l\). The object is held so that \(CA\) makes an angle \(\alpha\) with the downward vertical and is released from rest.

1. Given that \(\cos \alpha = \frac{1}{6}\), find the greatest speed achieved by the centre of the sphere in the subsequent motion. [6]

OR The times taken to run \(200\) metres at the beginning of the year and at the end of the year are recorded for each member of a large athletics club. The time taken, in seconds, at the beginning of the year is denoted by \(x\) and the time taken, in seconds, at the end of the year is denoted by \(y\). For a random sample of \(8\) members, the results are shown in the following table.

Member	A	B	C	D	E	F	G	H
\(x\)	24.2	23.8	22.8	25.1	24.5	24.0	23.8	22.8
\(y\)	23.9	23.6	22.8	24.5	24.2	23.5	23.6	22.7

\([\Sigma x = 191, \quad \Sigma x^2 = 4564.46, \quad \Sigma y = 188.8, \quad \Sigma y^2 = 4458.4, \quad \Sigma xy = 4510.99.]\)

1. Find, showing all necessary working, the equation of the regression line of \(y\) on \(x\). [4]

The athletics coach believes that, on average, the time taken by an athlete to run \(200\) metres decreases between the beginning and the end of the year by more than \(0.2\) seconds.

1. Stating suitable hypotheses and assuming a normal distribution, test the coach's belief at the \(10\%\) significance level. [8]

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]

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]

\includegraphics{figure_3} A particle of mass 0.24 kg is attached to one end of a light inextensible string of length 2 m. The other end of the string is attached to a fixed point. The particle moves with constant speed in a horizontal circle. The string makes an angle \(\theta\) with the vertical (see diagram), and the tension in the string is \(T\) N. The acceleration of the particle has magnitude \(7.5 \text{ m s}^{-2}\).

1. Show that \(\tan \theta = 0.75\) and find the value of \(T\). [4]
2. Find the speed of the particle. [2]

A horizontal circular disc rotates with constant angular speed \(9 \text{ rad s}^{-1}\) about its centre \(O\). A particle of mass \(0.05 \text{ kg}\) is placed on the disc at a distance \(0.4 \text{ m}\) from \(O\). The particle moves with the disc and no sliding takes place. Calculate the magnitude of the resultant force exerted on the particle by the disc. [3]

\includegraphics{figure_3} Particles \(P\) and \(Q\) have masses \(0.8\) kg and \(0.4\) kg respectively. \(P\) is attached to a fixed point \(A\) by a light inextensible string which is inclined at an angle \(\alpha°\) to the vertical. \(Q\) is attached to a fixed point \(B\), which is vertically below \(A\), by a light inextensible string of length \(0.3\) m. The string \(BQ\) is horizontal. \(P\) and \(Q\) are joined to each other by a light inextensible string which is vertical. The particles rotate in horizontal circles of radius \(0.3\) m about the axis through \(A\) and \(B\) with constant angular speed \(5\) rad s\(^{-1}\) (see diagram).

1. By considering the motion of \(Q\), find the tensions in the strings \(PQ\) and \(BQ\). [3]
2. Find the tension in the string \(AP\) and the value of \(\alpha\). [5]

A smooth bead \(B\) of mass 0.3 kg is threaded on a light inextensible string of length 0.9 m. One end of the string is attached to a fixed point \(A\), and the other end of the string is attached to a fixed point \(C\) which is vertically below \(A\). The tension in the string is 7 N, and the bead rotates with angular speed \(ω\) rad s\(^{-1}\) in a horizontal circle about the vertical axis through \(A\) and \(C\).

1. Given that \(B\) moves in a circle with centre \(C\) and radius 0.2 m, calculate \(ω\), and hence find the kinetic energy of \(B\). [5]
2. Given instead that angle \(ABC = 90°\), and that \(AB\) makes an angle \(\tan^{-1}(\frac{4}{3})\) with the vertical, calculate \(T\) and \(ω\). [6]

\(ABC\) is a uniform semicircular arc with diameter \(AC = 0.5\) m. The arc rotates about a fixed axis through \(A\) and \(C\) with angular speed \(2.4\) rad s\(^{-1}\). Calculate the speed of the centre of mass of the arc. [3]

\includegraphics{figure_4} A particle \(P\) is moving inside a smooth hollow cone which has its vertex downwards and its axis vertical, and whose semi-vertical angle is \(45°\). A light inextensible string parallel to the surface of the cone connects \(P\) to the vertex. \(P\) moves with constant angular speed in a horizontal circle of radius \(0.67\) m (see diagram). The tension in the string is equal to the weight of \(P\). Calculate the angular speed of \(P\). [6]

\includegraphics{figure_4} One end of a light inextensible string is attached to a fixed point \(A\). The string passes through a smooth bead \(B\) of mass \(0.3\,\text{kg}\) and the other end of the string is attached to a fixed point \(C\) vertically below \(A\). The bead \(B\) moves with constant speed in a horizontal circle of radius \(0.6\,\text{m}\) which has its centre between \(A\) and \(C\). The string makes an angle of \(30°\) with the vertical at \(A\) and an angle of \(45°\) with the vertical at \(C\) (see diagram).

1. Calculate the speed of \(B\). [5]

The lower end of the string is detached from \(C\), and \(B\) is now attached to this end of the string. The other end of the string remains attached to \(A\). The bead is set in motion so that it moves with angular speed \(3\,\text{rad s}^{-1}\) in a horizontal circle which has its centre vertically below \(A\).

1. Calculate the tension in the string. [3]