6.05e Radial/tangential acceleration

2 \includegraphics[max width=\textwidth, alt={}, center]{c397fca5-e7e8-4f3d-b519-cd92a983ebcc-02_810_743_831_644} 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 \(4.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). When \(O P\) makes an angle \(\theta\) with the upward vertical the speed of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) (see diagram).

1. Show that \(v ^ { 2 } = 33.32 - 15.68 \cos \theta\).
2. Prove that the bead is never at rest.
3. Find the maximum value of \(v\).
4. Write down the dimension of density. The workings of an oil pump consist of a right, solid cylinder which is partially submerged in oil. The cylinder is free to oscillate along its central axis which is vertical. If the base area of the pump is \(0.4 \mathrm {~m} ^ { 2 }\) and the density of the oil is \(920 \mathrm {~kg} \mathrm {~m} ^ { - 3 }\) then the period of oscillation of the pump is 0.7 s .
   A student assumes that the period of oscillation of the pump is dependent only on the density of the oil, \(\rho\), the acceleration due to gravity, \(g\), and the surface area, \(A\), of the circular base of the pump. The student attempts to test this assumption by stating that the period of oscillation, \(T\), is given by \(T = C \rho ^ { \alpha } g ^ { \beta } A ^ { \gamma }\) where \(C\) is a dimensionless constant.
5. Use dimensional analysis to find the values of \(\alpha , \beta\) and \(\gamma\).
6. Hence give the value of \(C\) to 3 significant figures.
7. Comment, with justification, on the assumption made by the student that the formula for the period of oscillation of the pump was dependent on only \(\rho , g\) and \(A\). A car of mass 1250 kg experiences a resistance to its motion of magnitude \(k v ^ { 2 } \mathrm {~N}\), where \(k\) is a constant and \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the car's speed. The car travels in a straight line along a horizontal road with its engine working at a constant rate of \(P \mathrm {~W}\). At a point \(A\) on the road the car's speed is \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and it has an acceleration of magnitude \(0.54 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). At a point \(B\) on the road the car's speed is \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and it has an acceleration of magnitude \(0.3 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
8. Find the values of \(k\) and \(P\). The power is increased to 15 kW .
9. Calculate the maximum steady speed of the car on a straight horizontal road.

7 A hollow cylinder, of internal radius 4 m , is fixed so that its axis is horizontal. The point \(O\) is on this axis. A particle, of mass 6 kg , is set in motion so that it moves on the smooth inner surface of the cylinder in a vertical circle about \(O\). Its speed at the point \(A\), which is vertically below \(O\), is \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). \includegraphics[max width=\textwidth, alt={}, center]{851cb2a3-5bc8-4af9-b1fc-a143d37beebe-5_746_739_504_662} When the particle is at the point \(B\), at a height of 2 m above \(A\), find:

1. its speed;
2. the normal reaction between the cylinder and the particle.

7 In crazy golf, a golf ball is hit so that it starts to move in a vertical circle on the inside of a smooth cylinder. Model the golf ball as a particle, \(P\), of mass \(m\). The circular path of the golf ball has radius \(a\) and centre \(O\). At time \(t\), the angle between \(O P\) and the horizontal is \(\theta\), as shown in the diagram. The golf ball has speed \(u\) at the lowest point of its circular path. \includegraphics[max width=\textwidth, alt={}, center]{9cfa110c-ee11-447a-b21a-3f436432e27d-6_739_742_719_641}

1. Show that, while the golf ball is in contact with the cylinder, the reaction of the cylinder on the golf ball is $$\frac { m u ^ { 2 } } { a } - 3 m g \sin \theta - 2 m g$$
2. Given that \(u = \sqrt { 3 a g }\), the golf ball will not complete a vertical circle inside the cylinder. Find the angle which \(O P\) makes with the horizontal when the golf ball leaves the surface of the cylinder.
   (4 marks)

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.

10 A particle \(P\) is attached to one end of a light inextensible string of length 1.4 m . The other end of the string is fixed to the ceiling at \(C\). The angle between \(C P\) and the vertical is \(\theta\) radians. The particle is held with the string taut with \(\theta = 0.3\) and is then released.

1. (a) Show that the motion of the system is approximately simple harmonic, and state its period.
   (b) Hence find an approximation for the speed of \(P\) when \(\theta = 0.2\).
2. Find the speed of \(P\) when \(\theta = 0.2\) using an energy method, and hence find the percentage error in the answer to part (i) (b).

6 A simple pendulum consists of a light inextensible string of length 1.5 m with a small bob of mass 0.2 kg at one end. When suspended from a fixed point and hanging at rest under gravity, the bob is given a horizontal speed of \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and it comes instantaneously to rest when the string makes an angle of 0.1 rad with the vertical. At time \(t\) seconds after projection the string makes an angle \(\theta\) with the vertical.

1. Show that, neglecting air resistance, $$\left( \frac { \mathrm { d } \theta } { \mathrm {~d} t } \right) ^ { 2 } = \frac { 40 } { 3 } \{ \cos \theta - \cos ( 0.1 ) \}$$
2. Find, correct to 2 significant figures,
   1. the value of \(u\),
   2. the tension in the string when \(\theta = 0.05 \mathrm { rad }\).
   3. By differentiating the above equation for \(\left( \frac { \mathrm { d } \theta } { \mathrm { d } t } \right) ^ { 2 }\), or otherwise, show that the motion of the bob can be modelled approximately by simple harmonic motion.
   4. Hence find the value of \(t\) at which the bob first comes instantaneously to rest.

\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]

\includegraphics{figure_5} A particle of mass \(m\) is attached to one end of a light inextensible string of length \(a\). The other end of the string is attached to a fixed point \(O\). The particle is moving in complete vertical circles with the string taut. When the particle is at the point \(P\), where \(OP\) makes an angle \(\alpha\) with the upward vertical through \(O\), its speed is \(u\). When the particle is at the point \(Q\), where angle \(QOP = 90°\), its speed is \(v\) (see diagram). It is given that \(\cos \alpha = \frac{4}{5}\).

1. Show that \(v^2 = u^2 + \frac{14}{5}ag\). [2]

The tension in the string when the particle is at \(Q\) is twice the tension in the string when the particle is at \(P\).

1. Obtain another equation relating \(u^2\), \(v^2\), \(a\) and \(g\), and hence find \(u\) in terms of \(a\) and \(g\). [5]
2. Find the least tension in the string during the motion. [3]

A particle \(P\) moves along an arc of a circle with centre \(O\) and radius 2 m. At time \(t\) seconds, the angle \(POA\) is \(\theta\), where \(\theta = 1 - \cos 2t\), and \(A\) is a fixed point on the arc of the circle.

1. Show that the magnitude of the radial component of the acceleration of \(P\) when \(t = \frac{1}{6}\pi\) is 6 m s\(^{-2}\). [2]
2. Find the magnitude of the transverse component of the acceleration of \(P\) when \(t = \frac{1}{6}\pi\). [2]

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]

A hollow cylinder of radius \(a\) is fixed with its axis horizontal. A particle \(P\), of mass \(m\), moves in part of a vertical circle of radius \(a\) and centre \(O\) on the smooth inner surface of the cylinder. The speed of \(P\) when it is at the lowest point \(A\) of its motion is \(\sqrt{\frac{7}{2}ga}\). The particle \(P\) loses contact with the surface of the cylinder when \(OP\) makes an angle \(\theta\) with the upward vertical through \(O\).

1. Show that \(\theta = 60°\). [5]
2. Show that in its subsequent motion \(P\) strikes the cylinder at the point \(A\). [5]

A particle \(Q\) of mass \(m\) is attached to a fixed point \(O\) by a light inextensible string of length \(a\). The particle moves in complete vertical circles about \(O\). The points \(A\) and \(B\) are on the path of \(Q\) with \(AB\) a diameter of the circle. \(OA\) makes an angle of \(60°\) with the downward vertical through \(O\) and \(OB\) makes an angle of \(60°\) with the upward vertical through \(O\). The speed of \(Q\) when it is at \(A\) is \(2\sqrt{ag}\). Given that \(T_A\) and \(T_B\) are the tensions in the string at \(A\) and \(B\) respectively, find the ratio \(T_A : T_B\). [6]

A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string of length \(a\). The other end of the string is attached to a fixed point \(O\). The particle \(P\) is held at the point \(A\), where \(OA\) makes an angle \(\theta\) with the downward vertical through \(O\), and with the string taut. The particle \(P\) is projected perpendicular to \(OA\) in an upwards direction with speed \(u\). It then starts to move along a circular path in a vertical plane. The string goes slack when \(P\) is at \(B\), where angle \(AOB\) is \(90°\) and the speed of \(P\) is \(\sqrt{\frac{1}{3}ag}\).

1. Find the value of \(\sin\theta\). [2]
2. Find, in terms of \(m\) and \(g\), the tension in the string when \(P\) is at \(A\). [5]

A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string of length \(a\). The other end of the string is attached to a fixed point \(O\). The particle \(P\) is held at the point \(A\), where \(OA\) makes an angle \(\alpha\) with the downward vertical through \(O\), and with the string taut. The particle \(P\) is projected perpendicular to \(OA\) in an upwards direction with speed \(\sqrt{3ag}\). It then starts to move along a circular path in a vertical plane. The string goes slack when \(P\) is at \(B\), where \(OB\) makes an angle \(\theta\) with the upward vertical. Given that \(\cos \alpha = \frac{3}{5}\), find the value of \(\cos \theta\). [4]

A particle \(P\), of mass \(m\), is attached to one end of a light inextensible string of length \(a\). The other end of the string is attached to a fixed point \(O\). The particle \(P\) moves in complete vertical circles about \(O\) with the string taut. The points \(A\) and \(B\) are on the path of \(P\) with \(AB\) a diameter of the circle. \(OA\) makes an angle \(\theta\) with the downward vertical through \(O\) and \(OB\) makes an angle \(\theta\) with the upward vertical through \(O\). The speed of \(P\) when it is at \(A\) is \(\sqrt{5ag}\). The ratio of the tension in the string when \(P\) is at \(A\) to the tension in the string when \(P\) is at \(B\) is \(9 : 5\).

1. Find the value of \(\cos \theta\). [6]
2. Find, in terms of \(a\) and \(g\), the greatest speed of \(P\) during its motion. [2]

A particle \(P\) of mass \(m\) is attached to one end of a light inextensible rod of length \(3a\). An identical particle \(Q\) is attached to the other end of the rod. The rod is smoothly pivoted at a point \(O\) on the rod, where \(OQ = x\). The system, of rod and particles, rotates about \(O\) in a vertical plane. At an instant when the rod is vertical, with \(P\) above \(Q\), the particle \(P\) is moving horizontally with speed \(u\). When the rod has turned through an angle of \(60°\) from the vertical, the speed of \(P\) is \(2\sqrt{ag}\), and the tensions in the two parts of the rod, \(OP\) and \(OQ\), have equal magnitudes.

1. Show that the speed of \(Q\) when the rod has turned through an angle of \(60°\) from the vertical is \(\frac{2x}{3a-x}\sqrt{ag}\). [2]
2. Find \(x\) in terms of \(a\). [5]
3. Find \(u\) in terms of \(a\) and \(g\). [4]

\includegraphics{figure_5} A bead of mass \(m\) moves on a smooth circular wire, with centre \(O\) and radius \(a\), in a vertical plane. The bead has speed \(v_A\) when it is at the point \(A\) where \(OA\) makes an angle \(\alpha\) with the downward vertical through \(O\), and \(\cos \alpha = \frac{2}{3}\). Subsequently the bead has speed \(v_B\) at the point \(B\), where \(OB\) makes an angle \(\theta\) with the upward vertical through \(O\). Angle \(AOB\) is a right angle (see diagram). The reaction of the wire on the bead at \(B\) is in the direction \(OB\) and has magnitude equal to \(\frac{1}{6}\) of the magnitude of the reaction when the bead is at \(A\).

1. Find, in terms of \(m\) and \(g\), the magnitude of the reaction at \(B\). [6]
2. Given that \(v_A = \sqrt{kag}\), find the value of \(k\). [2]

\includegraphics{figure_6} A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string of length \(a\). The other end of the string is attached to a fixed point \(O\). The particle \(P\) is held with the string taut and the string makes an angle \(\theta\) with the downward vertical through \(O\). The particle is projected at right angles to the string with speed \(\frac{1}{3}\sqrt{10ag}\) and begins to move downwards along a circular path. When the string is vertical, it strikes a small smooth peg at the point \(A\) which is vertically below \(O\). The circular path and the point \(A\) are in the same vertical plane. After the string strikes the peg, the particle \(P\) begins to move in a vertical circle with centre \(A\). When the string makes an angle \(\theta\) with the upward vertical through \(A\) the string becomes slack (see diagram). The distance of \(A\) below \(O\) is \(\frac{5}{6}a\).

1. Find the value of \(\cos \theta\). [6]
2. Find the ratio of the tensions in the string immediately before and immediately after it strikes the peg. [4]

\includegraphics{figure_1} A smooth solid hemisphere of radius 0.5 m is fixed with its plane face on a horizontal floor. The plane face has centre \(O\) and the highest point of the surface of the hemisphere is \(A\). A particle \(P\) has mass 0.2 kg. The particle is projected horizontally with speed \(u\) m s\(^{-1}\) from \(A\) and leaves the hemisphere at the point \(B\), where \(OB\) makes an angle \(\theta\) with \(OA\), as shown in Figure 1. The point \(B\) is at a vertical distance of 0.1 m below the level of \(A\). The speed of \(P\) at \(B\) is \(v\) m s\(^{-1}\)

1. Show that \(v^2 = u^2 + 1.96\) [3]
2. Find the value of \(u\). [4]

The particle first strikes the floor at the point \(C\).

1. Find the length of \(OC\). [7]

\includegraphics{figure_5} A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string of length \(a\). The other end of the string is fixed at the point \(O\). The particle is initially held with \(OP\) horizontal and the string taut. It is then projected vertically upwards with speed \(u\), where \(u^2 = 5ag\). When \(OP\) has turned through an angle \(\theta\) the speed of \(P\) is \(v\) and the tension in the string is \(T\), as shown in Figure 5.

1. Find, in terms of \(a\), \(g\) and \(\theta\), an expression for \(v^2\). [3]
2. Find, in terms of \(m\), \(g\) and \(\theta\), an expression for \(T\). [4]
3. Prove that \(P\) moves in a complete circle. [3]
4. Find the maximum speed of \(P\). [2]

\includegraphics{figure_3} A particle 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 \(O\). The particle is hanging at the point \(A\), which is vertically below \(O\). It is projected horizontally with speed \(u\). When the particle is at the point \(P\), \(\angle AOP = \theta\), as shown in Fig. 3. The string oscillates through an angle \(\alpha\) on either side of \(OA\) where \(\cos \alpha = \frac{2}{3}\).

1. Find \(u\) in terms of \(g\) and \(l\). [4]

When \(\angle AOP = \theta\), the tension in the string is \(T\).

1. Show that \(T = \frac{mg}{3}(9\cos\theta - 4)\). [6]
2. Find the range of values of \(T\). [4]

\includegraphics{figure_2} A particle is at the highest point \(A\) on the outer surface of a fixed smooth sphere of radius \(a\) and centre \(O\). The lowest point \(B\) of the sphere is fixed to a horizontal plane. The particle is projected horizontally from \(A\) with speed \(u\), where \(u < \sqrt{ag}\). The particle leaves the sphere at the point \(C\), where \(OC\) makes an angle \(\theta\) with the upward vertical, as shown in Fig. 2.

1. Find an expression for \(\cos \theta\) in terms of \(u\), \(g\) and \(a\). [7]

The particle strikes the plane with speed \(\sqrt{\frac{9ag}{2}}\).

1. Find, to the nearest degree, the value of \(\theta\). [7]

One end of a light inextensible string of length \(l\) is attached to a particle \(P\) of mass \(m\). The other end is attached to a fixed point \(A\). The particle is hanging freely at rest with the string vertical when it is projected horizontally with speed \(\sqrt{\frac{5gl}{2}}\).

1. Find the speed of \(P\) when the string is horizontal. [4]

When the string is horizontal it comes into contact with a small smooth fixed peg which is at the point \(B\), where \(AB\) is horizontal, and \(AB < l\). Given that the particle then describes a complete semicircle with centre \(B\),

1. Find the least possible value of the length \(AB\). [9]

A particle \(P\) is free to move on the smooth inner surface of a fixed thin hollow sphere of internal radius \(a\) and centre \(O\). The particle passes through the lowest point of the spherical surface with speed \(U\). The particle loses contact with the surface when \(OP\) is inclined at an angle \(\alpha\) to the upward vertical.

1. Show that \(U^2 = ag(2 + 3\cos \alpha)\). [7]

The particle has speed \(W\) as it passes through the level of \(O\). Given that \(\cos \alpha = \frac{1}{\sqrt{3}}\),

1. show that \(W^2 = ag\sqrt{3}\). [5]

A fixed smooth sphere has centre \(O\) and radius \(a\). A particle \(P\) is placed on the surface of the sphere at the point \(A\), where \(OA\) makes an angle \(\alpha\) with the upward vertical through \(O\). The particle is released from rest at \(A\). When \(OP\) makes an angle \(\theta\) to the upward vertical through \(O\), \(P\) is on the surface of the sphere and the speed of \(P\) is \(v\). Given that \(\cos \alpha = \frac{3}{5}\)

1. show that $$v^2 = \frac{2ga}{5}(3 - 5\cos \theta)$$ [4]
2. find the speed of \(P\) at the instant when it loses contact with the sphere. [8]