6.05e Radial/tangential acceleration

Three forces represented by the vectors \(-4\mathbf{i} + \mathbf{j} + 2\mathbf{j}\) and \(k\mathbf{i} - 2\mathbf{j}\) act at the points with coordinates \((0, 0)\), \((3, 0)\) and \((0, 4)\) respectively.

1. Given that the three forces form a couple, find the value of \(k\). [2]
2. Find the magnitude and direction of the couple. [3]

\includegraphics{figure_9} A particle P of mass \(m\) is joined to a fixed point O by a light inextensible string of length \(l\). P is released from rest with the string taut and making an acute angle \(\alpha\) with the downward vertical, as shown in Fig. 9. At a time \(t\) after P is released the string makes an angle \(\theta\) with the downward vertical and the tension in the string is \(T\). Angles \(\alpha\) and \(\theta\) are measured in radians.

1. Show that $$\left(\frac{\mathrm{d}\theta}{\mathrm{d}t}\right)^2 = \frac{2g}{l}\cos\theta + k_1,$$ where \(k_1\) is a constant to be determined in terms of \(g\), \(l\) and \(\alpha\). [4]
2. Show that $$T = 3mg\cos\theta + k_2,$$ where \(k_2\) is a constant to be determined in terms of \(m\), \(g\) and \(\alpha\). [3]

It is given that \(\alpha\) is small enough for \(\alpha^2\) to be negligible.

1. Find, in terms of \(m\) and \(g\), the approximate tension in the string. [2]
2. Show that the motion of P is approximately simple harmonic. [3]

\includegraphics{figure_10} A hollow sphere has centre O and internal radius \(r\). A bowl is formed by removing part of the sphere. The bowl is fixed to a horizontal floor, with its circular rim horizontal and the centre of the rim vertically above O. The point A lies on the rim of the bowl such that AO makes an angle of \(30°\) with the horizontal (see diagram). A particle P of mass \(m\) is projected from A, with speed \(u\), where \(u > \sqrt{\frac{gr}{2}}\), in a direction perpendicular to AO and moves on the smooth inner surface of the bowl. The motion of P takes place in the vertical plane containing O and A. The particle P passes through a point B on the inner surface, where OB makes an acute angle \(\theta\) with the vertical.

1. Determine, in terms of \(m\), \(g\), \(u\), \(r\) and \(\theta\), the magnitude of the force exerted on P by the bowl when P is at B. [7]

The difference between the magnitudes of the force exerted on P by the bowl when P is at points A and B is \(4mg\).

1. Determine, in terms of \(r\), the vertical distance of B above the floor. [4]

It is given that when P leaves the inner surface of the bowl it does not fall back into the bowl.

1. Show that \(u^2 > 2gr\). [5]

A particle P of mass 1 kg is fixed to one end of a light inextensible string of length 0.5 m. The other end of the string is attached to a fixed point O, which is 1.75 m above a horizontal plane. P is held with the string horizontal and taut. P is then projected vertically downwards with a speed of \(3.2 \text{ m s}^{-1}\).

1. Find the tangential acceleration of P when OP makes an angle of \(20°\) with the horizontal. [2]

The string breaks when the tension in it is 32 N. At this point the angle between OP and the horizontal is \(\theta\).

1. Show that \(\theta = 23.1°\), correct to 1 decimal place. [5]

Particle P subsequently hits the plane at a point A.

1. Determine the speed of P when it arrives at A. [4]
2. Show that A is almost vertically below O. [5]

\includegraphics{figure_13} A conical shell, of semi-vertical angle \(\alpha\), is fixed with its axis vertical and its vertex V upwards. A light inextensible string passes through a small smooth hole at V and a particle P of mass 4 kg hangs in equilibrium at one end of the string. The other end of the string is attached to a particle Q of mass 25 kg which moves in a horizontal circle at constant angular speed \(2.8 \text{ rad s}^{-1}\) on the smooth outer surface of the shell at a vertical depth \(h\) m below V (see diagram).

1. Show that \(k_1 h \sin^2 \alpha + k_2 \cos^2 \alpha = k_3 \cos \alpha\), where \(k_1\), \(k_2\) and \(k_3\) are integers to be determined. [7]
2. Determine the greatest value of \(h\) for which Q remains in contact with the shell. [3]

\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]

11 Answer only one of the following two alternatives.
EITHER
A particle \(P\) of mass \(m\) is suspended from a fixed point by a light elastic string of natural length \(l\), and hangs in equilibrium. The particle is pulled vertically down to a position where the length of the string is \(\frac { 13 } { 7 } l\). The particle is released from rest in this position and reaches its greatest height when the length of the string is \(\frac { 11 } { 7 } l\).

1. Show that the modulus of elasticity of the string is \(\frac { 7 } { 5 } \mathrm { mg }\).
2. Show that \(P\) moves in simple harmonic motion about the equilibrium position and state the period of the motion.
3. Find the time after release when the speed of \(P\) is first equal to half of its maximum value.
   OR
   For a random sample of 12 observations of pairs of values \(( x , y )\), the equation of the regression line of \(y\) on \(x\) and the equation of the regression line of \(x\) on \(y\) are $$y = b x + 4.5 \quad \text { and } \quad x = a y + c$$ respectively, where \(a , b\) and \(c\) are constants. The product moment correlation coefficient for the sample is 0.6 .
4. Test, at the \(5 \%\) significance level, whether there is evidence of positive correlation between the variables.
5. Given that \(b - a = 0.5\), find the values of \(a\) and \(b\).
6. Given that the sum of the \(x\)-values in the sample data is 66, find the value of \(c\) and sketch the two regression lines on the same diagram. For each of the 12 pairs of values of \(( x , y )\) in the sample, another variable \(z\) is considered, where \(z = 5 y\).
7. State the coefficient of \(x\) in the equation of the regression line of \(z\) on \(x\) and find the value of the product moment correlation coefficient between \(x\) and \(z\), justifying your answer.