Problems

A car of mass \(m\) makes a journey of distance \(2d\) in a straight line. It experiences air resistance and rolling resistance so that the total resistance to motion when it is moving with speed \(v\) is \(Av^2 +R\), where \(A\) and \(R\) are constants. The car starts from rest and moves with constant acceleration \(a\) for a distance \(d\). Show that the work done by the engine for this half of the journey is \[ \int_0^d (ma+R+Av^2) \, \d x \] and that it can be written in the form \[ \int_0^w \frac {(ma+R+Av^2)v}a\; \d v \,, \] where \(w =\sqrt {2ad\,}\,\). For the second half of the journey, the acceleration of the car is \(-a\).

1. In the case \(R>ma\), show that the work done by the engine for the whole journey is \[ 2Aad^2 + 2Rd \,. \]
2. In the case \(ma-2Aad< R< ma\), show that at a certain speed the driving force required to maintain the constant acceleration falls to zero. Thereafter, the engine does no work (and the driver applies the brakes to maintain the constant acceleration). Show that the work done by the engine for the whole journey is \[ 2Aad^2 + 2 Rd + \frac{(ma-R)^2}{4Aa} \, .\]

A small bullet of mass \(m\) is fired into a block of wood of mass \(M\) which is at rest. The speed of the bullet on entering the block is \(u\). Its trajectory within the block is a horizontal straight line and the resistance to the bullet's motion is \(R\), which is constant.

1. The block is fixed. The bullet travels a distance \(a\) inside the block before coming to rest. Find an expression for \(a\) in terms of \(m\), \(u\) and \(R\).
2. Instead, the block is free to move on a smooth horizontal table. The bullet travels a distance \(b\) inside the block before coming to rest relative to the block, at which time the block has moved a distance \(c\) on the table. Find expressions for \(b\) and \(c\) in terms of \(M\), \(m\) and \(a\).

An equilateral triangle, comprising three light rods each of length \(\sqrt3a\), has a particle of mass \(m\) attached to each of its vertices. The triangle is suspended horizontally from a point vertically above its centre by three identical springs, so that the springs and rods form a tetrahedron. Each spring has natural length \(a\) and modulus of elasticity \(kmg\), and is light. Show that when the springs make an angle \(\theta\) with the horizontal the tension in each spring is \[ \frac{ kmg(1-\cos\theta)}{\cos\theta}\,. \] Given that the triangle is in equilibrium when \(\theta = \frac16 \pi\), show that \(k=4\sqrt3 +6\). The triangle is released from rest from the position at which \(\theta=\frac13\pi\). Show that when it passes through the equilibrium position its speed \(V\) satisfies \[ V^2 = \frac{4ag}3(6+\sqrt3)\,. \]

A train consists of an engine and \(n\) trucks. It is travelling along a straight horizontal section of track. The mass of the engine and of each truck is \(M\). The resistance to motion of the engine and of each truck is \(R\), which is constant. The maximum power at which the engine can work is \(P\). Obtain an expression for the acceleration of the train when its speed is \(v\) and the engine is working at maximum power. The train starts from rest with the engine working at maximum power. Obtain an expression for the time \(T\) taken to reach a given speed \(V\), and show that this speed is only achievable if \[ P>(n+1)RV\,. \]

1. In the case when \((n+1) RV/P\) is small, use the approximation \(\ln (1-x) \approx -x -\frac12 x^2\) (valid for small \( x \)) to obtain the approximation \[ PT\approx \tfrac 12 (n+1) MV^2\, \] and interpret this result.
2. In the general case, the distance moved from rest in time \(T\) is \(X\). {\em Write down}, with explanation, an equation relating \(P\), \(T\), \(X\), \(M\), \(V\), \(R\) and \(n\) and hence show that \[ X= \frac{2PT - (n+1)MV^2}{2(n+1)R} \,. \]

Two thin discs, each of radius \(r\) and mass \(m\), are held on a rough horizontal surface with their centres a distance \(6r\) apart. A thin light elastic band, of natural length \(2\pi r\) and modulus \(\dfrac{\pi mg}{12}\), is wrapped once round the discs, its straight sections being parallel. The contact between the elastic band and the discs is smooth. The coefficient of static friction between each disc and the horizontal surface is \(\mu\), and each disc experiences a force due to friction equal to \(\mu mg\) when it is sliding. The discs are released simultaneously. If the discs collide, they rebound and a half of their total kinetic energy is lost in the collision.

1. Show that the discs start sliding, but come to rest before colliding, if and only if \mbox{\(\frac23 <\mu <1\)}.
2. Show that, if the discs collide at least once, their total kinetic energy just before the first collision is \(\frac43 mgr(2-3\mu)\).
3. Show that if \(\frac 4 9 > \mu^2 >\frac{5}{27}\) the discs come to rest exactly once after the first collision.

A particle of unit mass is projected vertically upwards with speed \(u\). At height \(x\), while the particle is moving upwards, it is found to experience a total force \(F\), due to gravity and air resistance, given by \(F=\alpha \e^{-\beta x}\), where \(\alpha\) and \(\beta\) are positive constants. Calculate the energy expended in reaching this height. Show that \[ F= {\textstyle \frac12} \beta v^2+ \alpha - {\textstyle \frac12} \beta u^2 \;, \] where \(v\) is the speed of the particle, and explain why \( \alpha = \frac12 \beta u^2 +g\), where \(g\) is the acceleration due to gravity. Determine an expression, in terms of \(y\), \(g\) and \(\beta\), for the air resistance experienced by the particle on its downward journey when it is at a distance \(y\) below its highest point.

A small ball of mass \(m\) is suspended in equilibrium by a light elastic string of natural length \(l\) and modulus of elasticity \(\lambda.\) Show that the total length of the string in equilibrium is \(l(1+mg/\lambda).\) If the ball is now projected downwards from the equilibrium position with speed \(u_{0},\) show that the speed \(v\) of the ball at distance \(x\) below the equilibrium position is given by \[ v^{2}+\frac{\lambda}{lm}x^{2}=u_{0}^{2}. \] At distance \(h\), where \(\lambda h^{2} < lmu_{0}^{2},\) below the equilibrium position is a horizontal surface on which the ball bounces with a coefficient of restitution \(e\). Show that after one bounce the velocity \(u_{1}\) at \(x=0\) is given by \[ u_{1}^{2}=e^{2}u_{0}^{2}+\frac{\lambda}{lm}h^{2}(1-e^{2}), \] and that after the second bounce the velocity \(u_{2}\) at \(x=0\) is given by \[ u_{2}^{2}=e^{4}u_{0}^{2}+\frac{\lambda}{lm}h^{2}(1-e^{4}). \]

\(ABCD\) is a horizontal line with \(AB=CD=a\) and \(BC=6a\). There are fixed smooth pegs at \(B\) and \(C\). A uniform string of natural length \(2a\) and modulus of elasticity \(kmg\) is stretched from \(A\) to \(D\), passing over the pegs at \(B\) and \(C\). A particle of mass \(m\) is attached to the midpoint \(P\) of the string. When the system is in equilibrium, \(P\) is a distance \(a/4\) below \(BC\). Evaluate \(k\). The particle is pulled down to a point \(Q\), which is at a distance \(pa\) below the mid-point of \(BC\), and is released from rest. \(P\) rises to a point \(R\), which is at a distance \(3a\) above \(BC\). Show that \(2p^2-p-17=0\). Show also that the tension in the strings is less when the particle is at \(R\) than when the particle is at \(Q\).

A projectile of mass \(m\) is fired horizontally from a toy cannon of mass \(M\) which slides freely on a horizontal floor. The length of the barrel is \(l\) and the force exerted on the projectile has the constant value \(P\) for so long as the projectile remains in the barrel. Initially the cannon is at rest. Show that the projectile emerges from the barrel at a speed relative to the ground of \[ \sqrt{\frac{2PMl}{m(M+m)}}. \]

A regular tetrahedron \(ABCD\) of mass \(M\) is made of 6 identical uniform rigid rods, each of length \(2a.\) Four light elastic strings \(XA,XB,XC\) and \(XD\), each of natural length \(a\) and modulus of elasticity \(\lambda,\) are fastened together at \(X\), the other end of each string being attached to the corresponding vertex. Given that \(X\) lies at the centre of mass of the tetrahedron, find the tension in each string. The tetrahedron is at rest on a smooth horizontal table, with \(B,C\) and \(D\) touching the table, and the ends of the strings at \(X\) attached to a point \(O\) fixed in space. Initially the centre of mass of the tetrahedron coincides with \(O.\) Suddenly the string \(XA\) breaks, and the tetrahedron as a result rises vertically off the table. If the maximum height subsequently attained is such that \(BCD\) is level with the fixed point \(O,\) show that (to 2 significant figures) \[ \frac{Mg}{\lambda}=0.098. \]

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Solution:

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The distance of \(A\) to \(X\) is \(\frac34\) the distance from \(A\) to the centre base (\(d\)) The distance of \(C\) to the centre of the base (\(G\)) is \(\frac{2}{3}\) the height of \(BCD\) which is \(\frac{\sqrt{3}}{2} \cdot 2a = \sqrt{3} a\). Therefore we must have \((2a)^2 = d^2 + \frac43a^2 \Rightarrow d = \frac{2\sqrt{2}}{\sqrt{3}}a\) and so \(AX = \frac34 \frac{2\sqrt{2}}{\sqrt{3}}a = \sqrt{\frac32}a\) The tension in each string will be \(\lambda \left (\sqrt{\frac32}-1 \right)\). Considering the energy of the system, when the ABCD reaches it's maximum height, it's velocity will be \(0\). Therefore the only energies to consider are GPE and EPE. Assuming the table is \(0\), we initially have \(EPE\) of \begin{align*} 3 \cdot \frac12 \lambda \frac{(a(\sqrt{\frac32}-1))^2}{a} = \frac32 \lambda a\left (\frac52-2\sqrt{\frac32} \right ) \end{align*} When \(BCD\) is level with \(O\), the height is \(\frac{1}{\sqrt{6}}a\) and GPE of \(\frac{Mga}{\sqrt{6}}\) The \(EPE\) will be: \begin{align*} 3 \cdot \frac12 \lambda \frac{(a(\frac{2}{\sqrt{3}}-1))^2}{a} &= \frac32 \lambda a \left (\frac73 - \frac{4}{\sqrt{3}}\right ) \end{align*} So by conservation of energy: \begin{align*} && \frac32 \lambda a\left (\frac52-2\sqrt{\frac32} \right ) &= \frac{Mga}{\sqrt{6}} + \frac32 \lambda a \left (\frac73 - \frac{4}{\sqrt{3}}\right ) \\ \Rightarrow && \frac{Mg}{\lambda} &= \sqrt{6} \left (\frac32 \left (\frac52-2\sqrt{\frac32} \right ) - \frac32 \left (\frac73 - \frac{4}{\sqrt{3}}\right ) \right) \\ &&&= -9 + 6\sqrt{2}+\sqrt{\frac38} \\ &&&= 0.09765380\ldots \\ &&&= 0.098\, (2\text{ s.f}) \end{align*}

A piece of circus apparatus consists of a rigid uniform plank of mass 1000\(\,\)kg, suspended in a horizontal position by two equal light vertical ropes attached to the ends. The ropes each have natural length 10\(\,\)m and modulus of elasticity 490\(\,\)000 N. Initially the plank is hanging in equilibrium. Nellie, an elephant of mass 4000\(\,\)kg, lands in the middle of the plank while travelling vertically downwards at speed 5\(\,\)ms\(^{-1}.\) While carrying Nellie, the plank comes instantaneously to rest at a negligible height above the floor, and at this instant Nellie steps nimbly and gently off the plank onto the floor. Assuming that the plank remains horizontal, and the rope remain vertical, throughout the motion, find to three significant figures its initial height above the floor. During the motion after Nellie alights, do the ropes ever become slack? {[}Take \(g\) to be \(9.8\mbox{\,\ ms}^{-1}.\){]}