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

183 questions

Sort by: Default | Easiest first | Hardest first
Edexcel M5 2017 June Q4
16 marks Challenging +1.8
  1. A uniform lamina \(P Q R\) of mass \(m\) is in the shape of an isosceles triangle, with \(P Q = P R = 5 a\) and \(Q R = 6 a\). The midpoint of \(Q R\) is \(T\).
    1. Show, using integration, that the moment of inertia of the lamina about an axis which passes through \(P\) and is parallel to \(Q R\), is \(8 m a ^ { 2 }\).
    2. Show, using integration, that the moment of inertia of the lamina about an axis which passes through \(P\) and \(T\), is \(1.5 m a ^ { 2 }\).
      [0pt] [You may assume without proof that the moment of inertia of a uniform rod, of mass \(m\) and length \(2 l\), about an axis perpendicular to the rod through its midpoint is \(\frac { 1 } { 3 } m l ^ { 2 }\) ]
      (4)
    The lamina is now free to rotate in a vertical plane about a fixed smooth horizontal axis \(A\) which passes through \(P\) and is perpendicular to the plane of the lamina. The lamina makes small oscillations about its position of stable equilibrium.
  2. By writing down an equation of rotational motion for the lamina as it rotates about \(A\), find the approximate period of these small oscillations.
Edexcel M5 2017 June Q5
15 marks Challenging +1.2
  1. A uniform rod \(A B\), of mass \(M\) and length \(2 L\), is free to rotate in a vertical plane about a smooth fixed horizontal axis through \(A\). The rod is hanging vertically at rest, with \(B\) below \(A\), when it is struck at its midpoint by a particle of mass \(\frac { 1 } { 2 } M\). Immediately before this impact, the particle is moving with speed \(u\), in a direction which is horizontal and perpendicular to the axis. The particle is brought to rest by the impact and immediately after the impact the rod moves with angular speed \(\omega\).
    1. Show that \(\omega = \frac { 3 u } { 8 L }\)
    Immediately after the impact, the magnitude of the vertical component of the force exerted on the \(\operatorname { rod }\) at \(A\) by the axis is \(\frac { 3 M g } { 2 }\)
  2. Find \(u\) in terms of \(L\) and \(g\).
  3. Show that the magnitude of the horizontal component of the force exerted on the rod at \(A\) by the axis, immediately after the impact, is zero. The rod first comes to instantaneous rest after it has turned through an angle \(\alpha\).
  4. Find the size of \(\alpha\). \includegraphics[max width=\textwidth, alt={}, center]{3ce3d486-0c4d-4d30-be86-e175b303fda8-19_56_58_2631_1875}
OCR MEI Further Mechanics B AS 2022 June Q1
9 marks Standard +0.3
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.
    1. Determine the value of \(T\).
    2. Hence find the tangential acceleration of the ring at this time.
OCR MEI Further Mechanics B AS 2021 November Q5
12 marks Moderate -0.3
5 On a fairground ride, the centre of a horizontal circular frame is attached to the top of a vertical pole, OP . When the frame and pole rotate, OP remains vertical and the frame remains horizontal. Chairs of mass 10 kg are attached to the frame by means of chains of length 2.5 m . The chains are modelled as being both light and inextensible. A side view of the situation when the ride is stationary is shown in Fig. 5. A chain fixed to point A on the circular frame supports a chair. The distance OA is 2 m . \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{37798594-8cb0-48aa-8401-090f09e25dff-5_839_1074_641_240} \captionsetup{labelformat=empty} \caption{Fig. 5}
\end{figure} A child of mass 40 kg sits in a chair and, after a short time, the ride is rotating at a steady angular speed of \(\omega\) radians per second, with the chain inclined at an angle of \(50 ^ { \circ }\) to the downward vertical. The motion of the child and chair is in a horizontal circle.
  1. Draw a sketch showing the forces acting on the chair when the ride is moving at this angular speed.
  2. - Determine the tension in the chain.
    On another occasion, a man of mass 90 kg sits in the chair; after a short time, the ride is rotating in a horizontal circle at a steady speed of \(\omega\) radians per second, with the chain inclined at the same angle of \(50 ^ { \circ }\) to the downward vertical.
  3. Without any detailed calculations, explain how your answers to part (b) for the child would compare with those for the man.
  4. Explain why the chain is modelled as light.
  5. State two other modelling assumptions that were used in answering part (b).
OCR MEI Further Mechanics B AS Specimen Q2
6 marks Moderate -0.8
2 A smooth wire is bent to form a circle of radius 2.5 m ; the circle is in a horizontal plane. A small ring of mass 0.2 kg is travelling round the wire.
  1. At one instant the ring is travelling at an angular speed of 120 revolutions per minute.
    (A) Calculate the angular speed in radians per second.
    (B) Calculate the component towards the centre of the circle of the force exerted on the ring by the wire.
  2. Why must the contact between the wire and the ring be smooth if your answer to part (i) ( \(B\) ) is also the total horizontal component of the force exerted on the ring by the wire?
OCR MEI Further Mechanics Major 2021 November Q11
16 marks Challenging +1.2
11 Two small uniform smooth spheres A and B , of equal radius, have masses 4 kg and 3 kg respectively. The spheres are placed in a smooth horizontal circular groove. The coefficient of restitution between the spheres is \(e\), where \(e > \frac { 2 } { 5 }\). At a given instant B is at rest and A is set moving along the groove with speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\). It may be assumed that in the subsequent motion the two spheres do not leave the groove.
  1. Determine, in terms of \(e\) and \(V\), the speeds of A and B immediately after the first collision.
  2. Show that the arc through which A moves between the first and second collisions subtends an angle at the centre of the circular groove of $$\frac { 2 \pi ( 4 - 3 e ) } { 7 e } \text { radians. }$$
    1. Determine, in terms of \(e\) and \(V\), the speed of B immediately after the second collision.
    2. What can be said about the motion of A and B if the collisions between A and B are perfectly elastic?
WJEC Further Unit 3 2019 June Q5
8 marks Moderate -0.5
5. The diagram shows a fairground ride that consists of a number of seats suspended by chains that swing out as the centre rotates. \includegraphics[max width=\textwidth, alt={}, center]{b430aa50-27e3-46f7-afef-7b8e75d46e1f-4_711_718_466_678} When the ride rotates at a constant angular speed of \(\omega = 1.4 \mathrm { rads } ^ { - 1 }\), the seats move in a horizontal circle with each chain making an angle \(\theta\) with the vertical. Each of the seats and the chains may be modelled as light. Assume that all chains have the same length and are inextensible. When a man of mass 75 kg occupies a seat, the tension in the chain is \(490 \sqrt { 3 } \mathrm {~N}\).
  1. Show that \(\theta = 30 ^ { \circ }\).
  2. Calculate the length of each chain.
Edexcel FM2 AS 2018 June Q2
9 marks Standard +0.3
  1. A car moves round a bend which is banked at a constant angle of \(\theta ^ { \circ }\) to the horizontal.
When the car is travelling at a constant speed of \(80 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) there is no sideways frictional force on the car. The car is modelled as a particle moving in a horizontal circle of radius 500 m .
  1. Find the value of \(\theta\).
  2. Identify one limitation of this model. The speed of the car is increased so that it is now travelling at a constant speed of \(90 \mathrm { kmh } ^ { - 1 }\) The car is still modelled as a particle moving in a horizontal circle of radius 500 m .
  3. Describe the extra force that will now be acting on the car, stating the direction of this force.
    VILU SIHI NI IIIUM ION OCVGHV SIHILNI IMAM ION OOVJYV SIHI NI JIIYM ION OC
Edexcel FM2 AS 2019 June Q3
9 marks Challenging +1.2
  1. A light inextensible string has length \(8 a\). 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 \(B\), with \(A\) vertically above \(B\) and \(A B = 4 a\). A small ball of mass \(m\) is attached to a point \(P\) on the string, where \(A P = 5 a\).
The ball moves in a horizontal circle with constant speed \(v\), with both \(A P\) and \(B P\) taut.
The string will break if the tension in it exceeds \(\frac { 3 m g } { 2 }\) By modelling the ball as a particle and assuming the string does not break,
  1. show that \(\frac { 9 a g } { 4 } < v ^ { 2 } \leqslant \frac { 27 a g } { 4 }\)
  2. find the least possible time needed for the ball to make one complete revolution.
Edexcel FM2 AS 2020 June Q2
13 marks Standard +0.3
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{0567d068-e23c-446e-9e11-f0c292972093-06_531_837_258_632} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} One end of a string of length \(3 a\) is attached to a point \(A\) and the other end is attached to a point \(B\) on a smooth horizontal table. The point \(B\) is vertically below \(A\) with \(A B = a \sqrt { 3 }\) A small smooth bead, \(P\), of mass \(m\) is threaded on to the string. The bead \(P\) moves on the table in a horizontal circle, with centre \(B\), with constant speed \(U\). Both portions, \(A P\) and \(B P\), of the string are taut, as shown in Figure 2. The string is modelled as being light and inextensible and the bead is modelled as a particle.
  1. Show that \(A P = 2 a\)
  2. Find, in terms of \(m , U\) and \(a\), the tension in the string.
  3. Show that \(U ^ { 2 } < a g \sqrt { 3 }\)
  4. Describe what would happen if \(U ^ { 2 } > a g \sqrt { 3 }\)
  5. State briefly how the tension in the string would be affected if the string were not modelled as being light.
Edexcel FM2 AS 2021 June Q2
10 marks Challenging +1.2
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a7901165-1679-4d30-9444-0c27020e32ea-04_572_889_246_589} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A small smooth ring \(P\), of mass \(m\), is threaded onto a light inextensible string of length 4a. One end of the string is attached to a fixed point \(A\) on a smooth horizontal table. The other end of the string is attached to a fixed point \(B\) which is vertically above \(A\). The ring moves in a horizontal circle with centre \(A\) and radius \(a\), as shown in Figure 2. The ring moves with constant angular speed \(\sqrt { \frac { 2 g } { 3 a } }\) about \(A B\).
The string remains taut throughout the motion.
  1. Find, in terms of \(m\) and \(g\), the magnitude of the normal reaction between \(P\) and the table. The angular speed of \(P\) is now gradually increased.
  2. Find, in terms of \(a\) and \(g\), the angular speed of \(P\) at the instant when it loses contact with the table.
  3. Explain how you have used the fact that \(P\) is smooth.
Edexcel FM2 AS 2024 June Q2
10 marks Standard +0.3
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{fd8bc7b5-adee-4d67-b15d-571255b00b83-04_529_794_246_639} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A thin hollow hemisphere, with centre \(O\) and radius \(a\), is fixed with its axis vertical, as shown in Figure 2. A small ball \(B\) of mass \(m\) moves in a horizontal circle on the inner surface of the hemisphere. The circle has centre \(C\) and radius \(r\). The point \(C\) is vertically below \(O\) such that \(O C = h\). The ball moves with constant angular speed \(\omega\) The inner surface of the hemisphere is modelled as being smooth and \(B\) is modelled as a particle. Air resistance is modelled as being negligible.
  1. Show that \(\omega ^ { 2 } = \frac { g } { h }\) Given that the magnitude of the normal reaction between \(B\) and the surface of the hemisphere is \(3 m g\)
  2. find \(\omega\) in terms of \(g\) and \(a\).
  3. State how, apart from ignoring air resistance, you have used the fact that \(B\) is modelled as a particle.
Edexcel FM2 AS Specimen Q2
16 marks Challenging +1.2
  1. A light inextensible string has length 7a. 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 \(B\), with \(A\) vertically above \(B\) and \(A B = 5 a\). A particle of mass \(m\) is attached to a point \(P\) on the string where \(A P = 4 a\). The particle moves in a horizontal circle with constant angular speed \(\omega\), with both \(A P\) and \(B P\) taut.
    1. Show that
      1. the tension in \(A P\) is \(\frac { 4 m } { 25 } \left( 9 a \omega ^ { 2 } + 5 g \right)\)
      2. the tension in \(B P\) is \(\frac { 3 m } { 25 } \left( 16 a \omega ^ { 2 } - 5 g \right)\).
    The string will break if the tension in it reaches a magnitude of \(4 m g\).
    The time for the particle to make one revolution is \(S\).
  2. Show that $$3 \pi \sqrt { \frac { a } { 5 g } } < S < 8 \pi \sqrt { \frac { a } { 5 g } }$$
  3. State how in your calculations you have used the assumption that the string is light.
Edexcel FM2 2022 June Q4
8 marks Standard +0.8
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{1f39620e-c10f-4344-89f1-626fff36d187-12_640_645_258_699} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A small smooth ring \(R\) of mass \(m\) is threaded onto a light inextensible string. One end of the string is attached to a fixed point \(A\) and the other end of the string is attached to the fixed point \(B\) such that \(B\) is vertically above \(A\) and \(A B = 6 a\) The ring moves with constant angular speed \(\omega\) in a horizontal circle with centre \(A\). The string is taut and \(B R\) makes a constant angle \(\theta\) with the downward vertical, as shown in Figure 2. The ring is modelled as a particle.
Given that \(\tan \theta = \frac { 8 } { 15 }\)
  1. find, in terms of \(m\) and \(g\), the magnitude of the tension in the string,
  2. find \(\omega\) in terms of \(a\) and \(g\)
Edexcel FM2 2023 June Q6
9 marks Standard +0.8
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3b070338-1de4-4c33-be29-d37ac06c9fed-20_611_782_210_660} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} A hollow right circular cone, of internal base radius 0.6 m and height 0.8 m , is fixed with its axis vertical and its vertex \(V\) pointing downwards, as shown in Figure 4. A particle \(P\) of mass \(m \mathrm {~kg}\) moves in a horizontal circle of radius 0.5 m on the rough inner surface of the cone. The particle \(P\) moves with constant angular speed \(\omega\) rads \(^ { - 1 }\) The coefficient of friction between the particle \(P\) and the inner surface of the cone is 0.25 Find the greatest possible value of \(\omega\)
Edexcel FM2 2024 June Q3
9 marks Standard +0.3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c14975b7-6afa-44ce-beab-1cba2e82b249-10_433_753_246_657} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a hemispherical bowl of internal radius \(10 d\) that is fixed with its circular rim horizontal. The centre of the circular rim is at the point \(O\).
A particle \(P\) moves with constant angular speed on the smooth inner surface of the bowl. The particle \(P\) moves in a horizontal circle with radius \(8 d\) and centre \(C\).
  1. Find, in terms of \(g\), the exact magnitude of the acceleration of \(P\). The time for \(P\) to complete one revolution is \(T\).
  2. Find \(T\) in terms of \(d\) and \(g\).
Edexcel FM2 Specimen Q2
8 marks Challenging +1.2
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f06704e5-454c-41c1-9577-b1210f60480d-04_655_643_207_639} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A hollow right circular cone, of base diameter \(4 a\) and height \(4 a\) is fixed with its axis vertical and vertex \(V\) downwards, as shown in Figure 1. A particle of mass \(m\) moves in a horizontal circle with centre \(C\) on the rough inner surface of the cone with constant angular speed \(\omega\). The height of \(C\) above \(V\) is \(3 a\).
The coefficient of friction between the particle and the inner surface of the cone is \(\frac { 1 } { 4 }\). Find, in terms of \(a\) and \(g\), the greatest possible value of \(\omega\).
CAIE M2 2010 June Q5
8 marks Standard +0.8
  1. It is given that when the ball moves with speed \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) the tension in the string \(Q B\) is three times the tension in the string \(P B\). Calculate the radius of the circle. The ball now moves along this circular path with the minimum possible speed.
  2. State the tension in the string \(P B\) in this case, and find the speed of the ball.
CAIE M2 2019 March Q6
8 marks Challenging +1.2
  1. Find, in terms of \(r\), the distance of the centre of mass of the prism from the centre of the cylinder.
    \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{b8e52188-f9a6-46fc-90bf-97965c6dd324-11_633_729_258_708} \captionsetup{labelformat=empty} \caption{Fig. 2}
    \end{figure} The prism has weight \(W \mathrm {~N}\) and is placed with its curved surface on a rough horizontal plane. The axis of symmetry of the cross-section makes an angle of \(30 ^ { \circ }\) with the vertical. A horizontal force of magnitude \(P \mathrm {~N}\) acting in the plane of the cross-section through the centre of mass is applied to the cylinder at the highest point of this cross-section (see Fig. 2). The prism rests in limiting equilibrium.
  2. Find the coefficient of friction between the prism and the plane.
CAIE M2 2008 November Q4
7 marks Standard +0.3
  1. the base of the cylinder,
  2. the curved surface of the cylinder.
    (ii) \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{5109244c-3062-4f5f-9277-fc6b5b28f2d4-3_348_745_1183_740} \captionsetup{labelformat=empty} \caption{Fig. 2}
    \end{figure} Sphere \(A\) is now attached to one end of a light inextensible string. The string passes through a small smooth hole in the middle of the base of the cylinder. Another small sphere \(B\), of mass 0.25 kg , is attached to the other end of the string. \(B\) hangs in equilibrium below the hole while \(A\) is moving in a horizontal circle of radius 0.2 m (see Fig. 2). Find the angular speed of \(A\).
OCR M2 2008 January Q6
11 marks Standard +0.3
  1. Show that the tension in the string is 4.16 N , correct to 3 significant figures.
  2. Calculate \(\omega\).
    (ii) \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{982647bd-8514-40cf-b4ee-674f51df32c5-3_510_417_1238_904} \captionsetup{labelformat=empty} \caption{Fig. 2}
    \end{figure} The lower part of the string is now attached to a point \(R\), vertically below \(P\). \(P B\) makes an angle \(30 ^ { \circ }\) with the vertical and \(R B\) makes an angle \(60 ^ { \circ }\) with the vertical. The bead \(B\) now moves in a horizontal circle of radius 1.5 m with constant speed \(v _ { \mathrm { m } } \mathrm { m } ^ { - 1 }\) (see Fig. 2).
    1. Calculate the tension in the string.
    2. Calculate \(v\).
OCR M2 2006 June Q6
11 marks Standard +0.3
  1. Calculate the tension in the string and hence find the angular speed of \(Q\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{d6d87705-be4b-407d-b699-69fb441d88a7-4_489_1358_1286_392} \captionsetup{labelformat=empty} \caption{Fig. 2}
    \end{figure} The particle \(Q\) on the plane is now fixed to a point 0.2 m from the hole at \(A\) and the particle \(P\) rotates in a horizontal circle of radius 0.2 m (see Fig. 2).
  2. Calculate the tension in the string.
  3. Calculate the speed of \(P\).
OCR M4 2008 June Q6
15 marks Challenging +1.3
  1. Show that the moment of inertia of the lamina about the axis through \(X\) is \(\frac { 4 } { 3 } m a ^ { 2 }\).
  2. At an instant when \(\cos \theta = \frac { 3 } { 5 }\), show that \(\omega ^ { 2 } = \frac { 6 g } { 5 a }\).
  3. At an instant when \(\cos \theta = \frac { 3 } { 5 }\), show that \(R = 0\), and given also that \(\sin \theta = \frac { 4 } { 5 }\) find \(S\) in terms of \(m\) and \(g\).
OCR Further Mechanics AS 2019 June Q5
14 marks Standard +0.8
  1. By considering forces on \(R\), express \(T _ { 2 }\) in terms of \(m _ { 2 }\).
  2. Show that
    1. \(T _ { 1 } = \frac { 49 } { 4 } \left( m _ { 1 } + m _ { 2 } \right)\),
    2. \(\omega ^ { 2 } = \frac { 49 \left( m _ { 1 } + 2 m _ { 2 } \right) } { 4 m _ { 1 } }\).
  3. Deduce that, in the case where \(m _ { 1 }\) is much bigger than \(m _ { 2 } , \omega \approx 3.5\). In a different case, where \(m _ { 1 } = 2.5\) and \(m _ { 2 } = 2.8 , P\) slows down. Eventually the system comes to rest with \(P\) and \(R\) hanging in equilibrium.
  4. Find the total energy lost by \(P\) and \(R\) as the angular velocity of \(P\) changes from the initial value of \(\omega \mathrm { rads } ^ { - 1 }\) to zero.
OCR FM1 AS 2017 December Q6
10 marks Standard +0.3
6 \includegraphics[max width=\textwidth, alt={}, center]{a1a43547-0a68-4346-884a-0c6d9302cf24-4_547_597_251_735} A particle of mass 0.2 kg is attached to one end of a light inextensible string. The other end of the string is attached to a fixed point \(O\) which is 1.8 m above a smooth horizontal table. The particle moves on the table in a circular path at constant speed with the string taut (see diagram). The particle has a speed of \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and its angular velocity is \(0.625 \mathrm { rad } \mathrm { s } ^ { - 1 }\).
  1. Show that the radius of the circular path is 0.8 m .
  2. Find the magnitude of the normal contact force between the particle and the table. The speed is changed to \(v \mathrm {~ms} ^ { - 1 }\). At this speed the particle is just about to lose contact with the table.
  3. Find the value of \(v\).