6.05d Variable speed circles: energy methods

\includegraphics{figure_7} A uniform rod \(AB\) has mass \(m\) and length \(6a\). It is free to rotate in a vertical plane about a smooth fixed horizontal axis passing through the point \(C\) on the rod, where \(AC = a\). The angle between \(AB\) and the upward vertical is \(\theta\), and the force acting on the rod at \(C\) has components \(R\) parallel to \(AB\) and \(S\) perpendicular to \(AB\) (see diagram). The rod is released from rest in the position where \(\theta = \frac{1}{4}\pi\). Air resistance may be neglected.

1. Find the angular acceleration of the rod in terms of \(a\), \(g\) and \(\theta\). [4]
2. Show that the angular speed of the rod is \(\sqrt{\frac{2g(1 - 2\cos\theta)}{7a}}\). [3]
3. Find \(R\) and \(S\) in terms of \(m\), \(g\) and \(\theta\). [6]
4. When \(\cos\theta = \frac{1}{3}\), show that the force acting on the rod at \(C\) is vertical, and find its magnitude. [4]

A uniform rod \(AB\) has mass \(2m\) and length \(4a\).

1. Show by integration that the moment of inertia of the rod about an axis perpendicular to the rod through \(A\) is \(\frac{32}{3}ma^2\). [4]

The rod is initially at rest with \(B\) vertically below \(A\) and it is free to rotate in a vertical plane about a smooth fixed horizontal axis through \(A\). A particle of mass \(m\) is moving horizontally in the plane in which the rod is free to rotate. The particle has speed \(v\), and strikes the rod at \(B\). In the subsequent motion the particle adheres to the rod and the combined rigid body \(Q\), consisting of the rod and the particle, starts to rotate.

1. Find, in terms of \(v\) and \(a\), the initial angular speed of \(Q\). [4]

At time \(t\) seconds the angle between \(Q\) and the downward vertical is \(\theta\) radians.

1. Show that \(\dot{\theta}^2 = k\frac{g}{a}(\cos \theta - 1) + \frac{9v^2}{400a^2}\), stating the value of the constant \(k\). [4]
2. Find, in terms of \(a\) and \(g\), the set of values of \(v^2\) for which \(Q\) makes complete revolutions. [2]

When \(Q\) is horizontal, the force exerted by the axis on \(Q\) has vertically upwards component \(R\).

1. Find \(R\) in terms of \(m\) and \(g\). [4]

\includegraphics{figure_2} A smooth wire is shaped into a circle of centre \(O\) and radius 0.8 m. The wire is fixed in a vertical plane. A small bead \(P\) of mass 0.03 kg is threaded on the wire and is projected along the wire from the highest point with a speed of \(4.2 \, \text{m s}^{-1}\). When \(OP\) makes an angle \(\theta\) with the upward vertical the speed of \(P\) is \(v \, \text{m s}^{-1}\) (see diagram).

1. Show that \(v^2 = 33.32 - 15.68\cos\theta\). [4]
2. Prove that the bead is never at rest. [1]
3. Find the maximum value of \(v\). [2]

\includegraphics{figure_10} A small toy car runs along a track in a vertical plane. The track consists of three sections: a curved section AB, a horizontal section BC which rests on the floor, and a circular section that starts at C with centre O and radius \(r\) m. The section BC is tangential to the curved section at B and tangential to the circular section at C, as shown in the diagram. The car, of mass \(m\) kg, is placed on the track at A, at a height \(h\) m above the floor, and released from rest. The car runs along the track from A to C and enters the circular section at C. It can be assumed that the track does not obstruct the car moving on to the circular section at C. The track is modelled as being smooth, and the car is modelled as a particle P.

1. Show that, while P remains in contact with the circular section of the track, the magnitude of the normal contact force between P and the circular section is $$mg\left(3\cos\theta - 2 + \frac{2h}{r}\right)\text{N},$$ where \(\theta\) is the angle between OC and OP. [7]
2. Hence determine, in terms of \(r\), the least possible value of \(h\) so that P can complete a vertical circle. [2]
3. Apart from not modelling the car as a particle, state one refinement that would make the model more realistic. [1]

\includegraphics{figure_10} Fig. 10 shows a small bead P of mass \(m\) which is threaded on a smooth thin wire. The wire is in the form of a circle of radius \(a\) and centre O. The wire is fixed in a vertical plane. The bead is initially at the lowest point A of the wire and is projected along the wire with a velocity which is just sufficient to carry it to the highest point on the wire. The angle between OP and the downward vertical is denoted by \(\theta\).

1. Determine the value of \(\theta\) when the magnitude of the reaction of the wire on the bead is \(\frac{7}{5}mg\). [7]
2. Show that the angular velocity of P when OP makes an angle \(\theta\) with the downward vertical is given by \(k\sqrt{\frac{g}{a}\cos\left(\frac{\theta}{2}\right)}\), stating the value of the constant \(k\). [4]
3. Hence determine, in terms of \(g\) and \(a\), the angular acceleration of P when \(\theta\) takes the value found in part (a). [3]

A fixed smooth sphere has centre O and radius \(a\). A particle P of mass \(m\) is placed at the highest point of the sphere and given an initial horizontal speed \(u\). For the first part of its motion, P remains in contact with the sphere and has speed \(v\) when OP makes an angle \(\theta\) with the upward vertical. This is shown in Fig. 4. \includegraphics{figure_4}

1. By considering the energy of P, show that \(v^2 = u^2 + 2ga(1 - \cos\theta)\). [2]
2. Show that the magnitude of the normal contact force between the sphere and particle P is $$mg(3\cos\theta - 2) - \frac{mv^2}{a}.$$ [2]

The particle loses contact with the sphere when \(\cos\theta = \frac{3}{4}\).

1. Find an expression for \(u\) in terms of \(a\) and \(g\). [2]

\includegraphics{figure_7} A smooth wire is shaped into a circle of radius 2.5 m which is fixed in a vertical plane with its centre at a point \(O\). A small bead \(B\) is threaded onto the wire. \(B\) is held with \(OB\) vertical and is then projected horizontally with an initial speed of \(8.4 \mathrm{ms}^{-1}\) (see diagram).

1. Find the speed of \(B\) at the instant when \(OB\) makes an angle of 0.8 radians with the downward vertical through \(O\). [3]
2. Determine whether \(B\) has sufficient energy to reach the point on the wire vertically above \(O\). [3]

A point \(O\) is situated a distance \(h\) above a smooth horizontal plane, and a particle \(A\) of mass \(m\) is attached to \(O\) by a light inextensible string of length \(h\). A particle \(B\) of mass \(2m\) is at rest on the plane, directly below \(O\), and is attached to a point \(C\) on the plane, where \(BC = l\), by a light inextensible string of length \(l\). \(A\) is released from rest with the string \(OA\) taut and making an acute angle \(\theta\) with the downward vertical (see diagram). \includegraphics{figure_8} \(A\) moves in a vertical plane perpendicular to \(CB\) and collides directly with \(B\). As a result of this collision, \(A\) is brought to rest and \(B\) moves on the plane in a horizontal circle with centre \(C\). After \(B\) has made one complete revolution the particles collide again.

1. Show that, on the next occasion that \(A\) comes to rest, the string \(OA\) makes an angle \(\phi\) with the downward vertical through \(O\), where \(\cos \phi = \frac{3 + \cos \theta}{4}\). [9]

\(A\) and \(B\) collide again when \(AO\) is next vertical.

1. Find the percentage of the original energy of the system that remains immediately after this collision. [5]
2. Explain why the total momentum of the particles immediately before the first collision is the same as the total momentum of the particles immediately after the second collision. [1]
3. Explain why the total momentum of the particles immediately before the first collision is different from the total momentum of the particles immediately after the third collision. [1]

One end of a light inextensible string of length \(l\) is attached to a fixed point \(O\). The other end is attached to a particle \(P\) of mass \(m\). The particle hangs at rest vertically below \(O\). The particle is then given a horizontal speed \(u\).

1. 1. Show that when \(OP\) has turned through an angle \(\theta\) the tension in the string is given by $$T = mg(3\cos \theta - 2) + \frac{mu^2}{l}$$ as long as the string remains taut. [5]
   2. Deduce that \(u^2 \geq 5gl\) in order for the particle to perform complete circles. [1]
2. 1. In the case \(u^2 = 3gl\), find the angle that \(OP\) makes with the downward vertical at \(O\) at the instant when the string becomes slack. [2]
   2. Describe the nature of the motion while the string is slack. [1]