Questions — Edexcel M3 (510 questions)

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Edexcel M3 Q4
11 marks Challenging +1.8
A particle \(P\) of mass \(m\) kg moves along a straight line under the action of a force of magnitude \(\frac{km}{x^2}\) N, where \(k\) is a constant, directed towards a fixed point \(O\) on the line, where \(OP = x\) m. \(P\) starts from rest at \(A\), at a distance \(a\) m from \(O\). When \(OP = x\) m, the speed of \(P\) is \(v\) ms\(^{-1}\).
  1. Show that \(v = \sqrt{\frac{2k(a-x)}{ax}}\). [6 marks]
\(B\) is the point half-way between \(O\) and \(A\). When \(k = \frac{1}{2}\) and \(a = 1\), the time taken by \(P\) to travel from \(A\) to \(B\) is \(T\) seconds Assuming the result that, for \(0 \leq x \leq 1\), \(\int \sqrt{\frac{x}{1-x}} dx = \arcsin(\sqrt{x}) - \sqrt{x(1-x^2)} + \text{constant}\),
  1. find the value of \(T\). [5 marks]
Edexcel M3 Q5
13 marks Standard +0.8
A car moves round a circular racing track of radius 100 m, which is banked at an angle of 4° to the horizontal.
  1. Show that when its speed is 8.28 ms\(^{-1}\), there is no sideways force acting on the car. [4 marks]
  2. When the speed of the car is 12.5 ms\(^{-1}\), find the smallest value of the coefficient of friction between the car and the track which will prevent side-slip. [9 marks]
Edexcel M3 Q6
13 marks Standard +0.8
The diagram shows a particle \(P\) of mass \(m\) kg moving on the inner surface of a smooth fixed hemispherical bowl of radius \(r\) m which is fixed with its axis vertical. \(P\) moves at a constant speed in a horizontal circle, at a depth \(h\) m below the top of the bowl. \includegraphics{figure_6}
  1. Show that the force \(R\) exerted on \(P\) by the bowl has magnitude \(\frac{mgr}{h}\) N. [4 marks]
  2. Find, in terms of \(g\), \(h\) and \(r\), the constant speed of \(P\). [4 marks]
The bowl is now inverted and \(P\) moves on the smooth outer surface at a height \(h\) above the plane face under the action of a force of magnitude \(mg\) applied tangentially as shown. The reaction of the surface of the sphere on \(P\) now has magnitude \(S\) N. \includegraphics{figure_6b}
  1. Given that \(r = 2h\), prove that \(S < \frac{1}{6}R\). [5 marks]
Edexcel M3 Q7
15 marks Challenging +1.8
A particle \(P\) of mass \(m\) kg is fixed to one end of a light elastic string of modulus \(mg\) N and natural length \(l\) m. The other end of the string is attached to a fixed point \(O\) on a rough horizontal table. Initially \(P\) is at rest in limiting equilibrium on the table at the point \(X\) where \(OX = \frac{5l}{4}\) m.
  1. Find the coefficient of friction between \(P\) and the table. [2 marks]
\(P\) is now given a small displacement \(x\) m horizontally along \(OX\), away from \(O\). While \(P\) is in motion, the frictional resistance remains constant at its limiting value.
  1. Show that as long as the string remains taut, \(P\) performs simple harmonic motion with \(X\) as the centre. [4 marks]
If \(P\) is held at the point where the extension in the string is \(l\) m and then released,
  1. show that the string becomes slack after a time \(\left(\frac{\pi}{2} + \arcsin\left(\frac{1}{3}\right)\right)\sqrt{\frac{l}{g}}\) s. [5 marks]
  2. Determine the speed of \(P\) when it reaches \(O\). [4 marks]
Edexcel M3 Q1
6 marks Moderate -0.3
A cyclist travels on a banked track inclined at \(8°\) to the horizontal. He moves in a horizontal circle of radius 10 m at a constant speed of \(v\) ms\(^{-1}\). If there is no sideways frictional force on the cycle, calculate the value of \(v\). [6 marks]
Edexcel M3 Q2
9 marks Standard +0.3
The figure shows a particle \(P\), of mass 0·8 kg, attached to the ends of two light elastic strings. \(AP\) has natural length 20 cm and modulus of elasticity \(\lambda\) N. \(BP\) has natural length 20 cm and modulus of elasticity \(\mu\) N. \(A\) and \(B\) are fixed to points on the same horizontal level so that \(AB = 50\) cm. When \(P\) is suspended in equilibrium, \(AP = 30\) cm and \(BP = 40\) cm. Calculate the values of \(\lambda\) and \(\mu\). \includegraphics{figure_2} [9 marks]
Edexcel M3 Q3
10 marks Challenging +1.8
Suraiya, whose mass is \(m\) kg, takes a running jump into a swimming pool so that she begins to swim in a straight line with speed 0·2 ms\(^{-1}\). She continues to move in the same straight line, the only force acting on her being a resistance of magnitude \(mv^2 \sin \left(\frac{t}{100}\right)\) N, where \(v\) ms\(^{-1}\) is her speed at time \(t\) seconds after entering the pool and \(0 \leq t \leq 50\pi\).
  1. Find an expression for \(v\) in terms of \(t\). [7 marks]
  2. Calculate her greatest and least speeds during her motion. [3 marks]
Edexcel M3 Q4
12 marks Challenging +1.2
A uniform lamina is in the shape of the region enclosed by the coordinate axes and the curve with equation \(y = 1 + \cos x\), as shown. \includegraphics{figure_4}
  1. Show by integration that the centre of mass of the lamina is at a distance \(\frac{\pi^2 - 4}{2\pi}\) from the \(y\)-axis. [9 marks]
Given that the centre of mass is at a distance 0·75 units from the \(x\)-axis, and that \(P\) is the point \((0, 2)\) and \(O\) is the origin \((0, 0)\),
  1. find, to the nearest degree, the angle between the line \(OP\) and the vertical when the lamina is freely suspended from \(P\). [3 marks]
Edexcel M3 Q5
12 marks Standard +0.8
A particle \(P\), of mass 0·5 kg, rests on the surface of a rough horizontal table. The coefficient of friction between \(P\) and the table is 0·5. \(P\) is connected to a particle \(Q\), of mass 0·2 kg, by a light inextensible string passing through a small smooth hole at a point \(O\) on the table, such that the distance \(OQ\) is 0·4 m. \(Q\) moves in a horizontal circle while \(P\) remains in limiting equilibrium. \includegraphics{figure_5}
  1. Calculate the angle \(\theta\) which \(OQ\) makes with the vertical. [4 marks]
  2. Show that the speed of \(Q\) is 1·33 ms\(^{-1}\). [3 marks]
The motion is altered so that \(Q\) hangs at rest below \(O\) and \(P\) moves in a horizontal circle on the table with speed 0·84 ms\(^{-1}\), at a constant distance \(r\) m from \(O\) but tending to slip away from \(O\).
  1. Find the value of \(r\). [5 marks]
Edexcel M3 Q6
12 marks Standard +0.3
The figure shows a swing consisting of two identical vertical light springs attached symmetrically to a light horizontal cross-bar and supported from a strong fixed horizontal beam. When a mass of 24 kg is placed at the mid-point of the cross-bar, both springs extend by 30 cm to the position \(A\), as shown. \includegraphics{figure_6} Each spring has natural length \(l\) m and modulus of elasticity \(\lambda\) N.
  1. Show that \(\lambda = 392l\). [2 marks]
The 24 kg mass is left on the bar and the bar is then displaced downwards by a further 20 cm.
  1. Prove that the system comprising the bar and the mass now performs simple harmonic motion with the centre of oscillation at the level \(A\). [5 marks]
  2. Calculate the number of oscillations made per second in this motion. [3 marks]
  3. Find the maximum acceleration which the mass experiences during the motion. [2 marks]