Mechanics 3 (M3): Further dynamicsEdexcel International A Level Further Maths: Topic test

1. 1
   A particle $P$ of mass $2$ kg moves along the positive $x$-axis. When $P$ is at displacement $x$ metres from the origin $O$, the only force acting on $P$ is a force of magnitude $\frac{8}{x^2}$ N acting in the direction of increasing $x$. When $x=1$, the speed of $P$ is $2$ m s$^{-1}$.

   (a)

   Find the acceleration of $P$ at displacement $x$.

   [1 mark]

   • A$\frac{8}{x^2}$ m s$^{-2}$
   • B$\frac{16}{x^2}$ m s$^{-2}$
   • C$\frac{4}{x^2}$ m s$^{-2}$
   • D$-\frac{4}{x^2}$ m s$^{-2}$

   (b)

   Which equation, relating $v$ and $x$, correctly describes the motion of $P$?

   [1 mark]

   • A$v\frac{\mathrm{d}v}{\mathrm{d}x}=\frac{4}{x^2}$
   • B$\frac{\mathrm{d}v}{\mathrm{d}x}=\frac{4}{x^2}$
   • C$v\frac{\mathrm{d}v}{\mathrm{d}x}=\frac{8}{x^2}$
   • D$v\frac{\mathrm{d}v}{\mathrm{d}x}=-\frac{4}{x^2}$

   (c)

   Find $v^2$ as a function of $x$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A particle $P$ moves in a straight line with simple harmonic motion about a fixed centre $O$. The period of the motion is $\frac{\pi}{2}$ s and the amplitude is $0.6$ m.

   (a)

   Find the angular frequency $\omega$ of the motion, in rad s$^{-1}$.

   [1 mark]

   • A$\frac14$
   • B$\frac2\pi$
   • C$2$
   • D$4$

   (b)

   Find the maximum speed of $P$, in m s$^{-1}$.

   [1 mark]

   • A$9.6$
   • B$2.4$
   • C$1.44$
   • D$0.15$

   (c)

   Find the speed of $P$ when it is $0.3$ m from $O$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A sledge of mass $50$ kg slides in a straight line along a horizontal path of snow. The only horizontal force acting on the sledge is a resistance of magnitude $5v$ N, where $v$ m s$^{-1}$ is the speed of the sledge at time $t$ seconds. At $t=0$, $v=8$.

   (a)

   Show that $v=8\mathrm{e}^{-\frac{t}{10}}$.

   [3 marks]

   (b)

   Find the distance the sledge travels while its speed falls from $8$ m s$^{-1}$ to $2$ m s$^{-1}$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A particle $P$ of mass $0.4$ kg lies on a smooth horizontal table. It is attached to the ends of two light elastic strings, each of natural length $0.5$ m and modulus of elasticity $20$ N. The other ends of the strings are fixed to points $A$ and $B$ on the table, where $AB=2$ m, and $P$ rests in equilibrium at the midpoint $M$ of $AB$. The particle is displaced a distance $x$ metres from $M$ towards $B$, along $AB$, with both strings taut.

   (a)

   Show that $P$ moves with simple harmonic motion, and find the period of the motion.

   [6 marks]

   (b)

   The particle is released from rest when $x=0.3$. Find (i) the maximum speed of $P$, (ii) the time taken for $P$ to reach the point where $x=0.15$ for the first time.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   A light spring of natural length $0.4$ m and modulus of elasticity $32$ N stands vertically with its lower end fixed to a horizontal floor. A block $B$ of mass $0.8$ kg, modelled as a particle, rests in equilibrium on the upper end of the spring. Take $g=9.8$ m s$^{-2}$.

   (a)

   Find the compression of the spring in equilibrium, in metres.

   [1 mark]

   • A$0.245$
   • B$0.098$
   • C$0.302$
   • D$0.6125$

   (b)

   $B$ is displaced vertically and oscillates with the spring always compressed. Find $\omega^2$ for the motion.

   [1 mark]

   • A$80$
   • B$10$
   • C$40$
   • D$100$

   (c)

   $B$ is pressed down a further $0.05$ m from its equilibrium position and released from rest. Show that $B$ remains in contact with the spring throughout the subsequent motion.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   A particle $P$ of mass $4$ kg moves in a straight line. At time $t$ seconds, $t\ge0$, the resultant force on $P$ is $16\cos2t$ N in the direction of motion. At $t=0$, $P$ is at rest at the point $O$.

   (a)

   Find the speed of $P$ when $t=\frac\pi4$, in m s$^{-1}$.

   [1 mark]

   • A$2$
   • B$8$
   • C$4$
   • D$\sqrt2$

   (b)

   Find the displacement of $P$ from $O$ when $t=\frac\pi6$, in metres.

   [1 mark]

   • A$1.5$
   • B$-0.5$
   • C$0.5$
   • D$1$

   (c)

   Find the first time after $t=0$ at which $P$ is instantaneously at rest.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   A particle $P$ moves in a straight line with simple harmonic motion about a fixed centre $O$. When $P$ is $0.6$ m from $O$ its speed is $4$ m s$^{-1}$, and when $P$ is $0.8$ m from $O$ its speed is $3$ m s$^{-1}$.

   (a)

   Find the amplitude of the motion and the angular frequency $\omega$.

   [3 marks]

   (b)

   Find the time taken for $P$ to travel directly from $O$ to the point $0.8$ m from $O$.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   A particle $P$ of mass $0.6$ kg is attached to one end of a light elastic string of natural length $0.75$ m and modulus of elasticity $45$ N. The other end of the string is attached to a fixed point $O$, and $P$ hangs in equilibrium vertically below $O$. Take $g=9.8$ m s$^{-2}$.

   (a)

   The particle is pulled vertically downwards and released, and the string remains taut. Show that $P$ moves with simple harmonic motion and find the period of the motion.

   [6 marks]

   (b)

   $P$ is pulled down a further $0.2$ m from the equilibrium position and released from rest. Find the greatest height of $P$ above the equilibrium position.

   [6 marks]

   Total for question 8: 12 marks

End of questions