Edexcel M3 (Mechanics 3)

1. \begin{figure}[h] \end{figure} A particle of mass 0.6 kg is attached to one end of a light elastic spring of natural length 1 m and modulus of elasticity 30 N . The other end of the spring is fixed to a point \(O\) which lies on a smooth plane inclined at an angle \(\alpha\) to the horizontal where \(\tan \alpha = \frac { 3 } { 4 }\) as shown in Figure 1. The particle is held at rest on the slope at a point 1.2 m from \(O\) down the line of greatest slope of the plane.

1. Find the tension in the spring.
2. Find the initial acceleration of the particle.

2. A particle \(P\) of mass 0.5 kg moves along the positive \(x\)-axis under the action of a single force directed away from the origin \(O\). When \(P\) is \(x\) metres from \(O\), the magnitude of the force is \(3 x ^ { \frac { 1 } { 2 } } \mathrm {~N}\) and \(P\) has a speed of \(v \mathrm {~ms} ^ { - 1 }\). Given that when \(x = 1 , P\) is moving away from \(O\) with speed \(2 \mathrm {~ms} ^ { - 1 }\),

1. find an expression for \(v ^ { 2 }\) in terms of \(x\),
2. show that when \(x = 4 , P\) has a speed of \(7.7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), correct to 1 decimal place.

3. A particle is performing simple harmonic motion along a straight line between the points \(A\) and \(B\) where \(A B = 8 \mathrm {~m}\). The period of the motion is 12 seconds.

1. Find the maximum speed of the particle in terms of \(\pi\). The points \(P\) and \(Q\) are on the line \(A B\) at distances of 3 m and 6 m respectively from \(A\).
2. Find, correct to 3 significant figures, the time it takes for the particle to travel directly from \(P\) to \(Q\).
   (6 marks)

4. Whilst in free-fall a parachutist falls vertically such that his velocity, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), when he is \(x\) metres below his initial position is given by $$v ^ { 2 } = k g \left( 1 - \mathrm { e } ^ { - \frac { 2 x } { k } } \right) ,$$ where \(k\) is a constant.
Given that he experiences an acceleration of \(\mathrm { f } \mathrm { m } \mathrm { s } ^ { - 2 }\),

1. show that \(f = g \mathrm { e } ^ { - \frac { 2 x } { k } }\). After falling a large distance, his velocity is constant at \(49 \mathrm {~ms} ^ { - 1 }\).
2. Find the value of \(k\).
3. Hence, express \(f\) in the form ( \(\lambda - \mu v ^ { 2 }\) ) where \(\lambda\) and \(\mu\) are constants which you should find.
   (4 marks)

5. \begin{figure}[h] \end{figure} A firework is modelled as a uniform solid formed by joining the plane surface of a right circular cone of height \(2 r\) and base radius \(r\), to one of the plane surfaces of a cylinder of height \(h\) and base radius \(r\) as shown in Figure 2. Using this model,

1. show that the distance of the centre of mass of the firework from its plane base is $$\frac { 3 h ^ { 2 } + 4 h r + 2 r ^ { 2 } } { 2 ( 3 h + 2 r ) }$$ The firework is to be launched from rough ground inclined at an angle \(\alpha\) to the horizontal. Given that the firework does not slip or topple and that \(h = 4 r\),
2. Find, correct to the nearest degree, the maximum value of \(\alpha\).

6. \begin{figure}[h] \end{figure} The two ends of a light inextensible string of length \(3 a\) are attached to fixed points \(Q\) and \(R\) which are a distance of \(a \sqrt { } 3\) apart with \(R\) vertically below \(Q\). A particle \(P\) of mass \(m\) is attached to the string at a distance of \(2 a\) from \(Q\).
\(P\) is given a horizontal speed, \(u\), such that it moves in a horizontal circle with both sections of the string taut as shown in Figure 3.

1. Show that \(\angle P R Q\) is a right angle.
2. Find \(\angle P Q R\) in degrees.
3. Find, in terms of \(a , g , m\) and \(u\), the tension in the section of string
   1. \(P Q\),
   2. \(P R\).
4. Show that \(u ^ { 2 } \geq \frac { g a } { \sqrt { 3 } }\).

7. A particle of mass 2 kg is attached to one end of a light elastic string of natural length 1 m and modulus of elasticity 50 N . The other end of the string is attached to a fixed point \(O\) on a rough horizontal plane and the coefficient of friction between the particle and the plane is \(\frac { 10 } { 49 }\). The particle is projected from \(O\) along the plane with an initial speed of \(5 \mathrm {~ms} ^ { - 1 }\).

1. Show that the greatest distance from \(O\) which the particle reaches is 1.84 m .
2. Find, correct to 2 significant figures, the speed at which the particle returns to \(O\).