Further Mechanics 2: Motion in a circleEdexcel A-Level Further Maths: Topic test

1. 1
   A toy car of mass $0.15$ kg is driven round a horizontal circular track of radius $0.8$ m at constant speed. The car completes $30$ revolutions per minute and is modelled as a particle.

   (a)

   What is the angular speed of the car?

   [1 mark]

   • A$0.5$ rad s$^{-1}$
   • B$\pi$ rad s$^{-1}$
   • C$\frac{\pi}{2}$ rad s$^{-1}$
   • D$60\pi$ rad s$^{-1}$

   (b)

   What is the speed of the car?

   [1 mark]

   • A$1.26$ m s$^{-1}$
   • B$7.90$ m s$^{-2}$
   • C$0.255$ m s$^{-1}$
   • D$2.51$ m s$^{-1}$

   (c)

   Find the magnitude of the resultant horizontal force acting on the car.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A particle $P$ of mass $0.3$ kg is attached 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 $O$. $P$ moves in a horizontal circle, with its centre vertically below $O$, with the string making a constant angle $\theta$ with the downward vertical, where $\cos\theta=0.8$. Take $g=9.8$ m s$^{-2}$.

   (a)

   What is the tension in the string?

   [1 mark]

   • A$3.68$ N
   • B$2.94$ N
   • C$4.90$ N
   • D$2.21$ N

   (b)

   What is the angular speed of $P$?

   [1 mark]

   • A$24.5$ rad s$^{-1}$
   • B$3.83$ rad s$^{-1}$
   • C$4.95$ rad s$^{-1}$
   • D$4.43$ rad s$^{-1}$

   (c)

   Find the speed of $P$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A small coin of mass $0.012$ kg lies on a rough horizontal turntable, at a distance $0.15$ m from the vertical axis of rotation. The turntable rotates at a constant angular speed. The coefficient of friction between the coin and the turntable is $0.4$. The coin is modelled as a particle.

   (a)

   Find the greatest angular speed of the turntable for which the coin does not slip.

   [3 marks]

   (b)

   The coin is moved to a new position on the turntable, which now rotates at $6$ rad s$^{-1}$. The coin is on the point of slipping. Find its distance from the axis and the magnitude of the frictional force on it.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A particle $P$ of mass $0.25$ kg is attached to one end of a light inextensible string of length $0.9$ m. The other end of the string is attached to a fixed point $O$. Model $P$ as a particle and take $g=9.8$ m s$^{-2}$.

   (a)

   $P$ moves in a horizontal circle with the string making a constant angle $\theta$ with the downward vertical, where $\cos\theta=0.8$. Find the tension in the string and the angular speed of $P$.

   [6 marks]

   (b)

   $P$ now hangs at rest vertically below $O$ and is projected horizontally with speed $7$ m s$^{-1}$. Show that $P$ moves in a complete vertical circle, and find the greatest and least values of the tension in the string.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   A car of mass $900$ kg crosses a humped bridge. The hump is modelled as an arc of a vertical circle of radius $50$ m, and the car is modelled as a particle. At the highest point of the bridge the car has speed $14$ m s$^{-1}$. Take $g=9.8$ m s$^{-2}$.

   (a)

   What is the magnitude of the resultant force on the car at the highest point, in N?

   [1 mark]

   • A$8820$
   • B$5292$
   • C$3528$
   • D$12348$

   (b)

   What is the normal reaction of the road on the car at the highest point, in N?

   [1 mark]

   • A$5292$
   • B$12348$
   • C$3528$
   • D$8820$

   (c)

   Find the greatest speed at which the car can cross the top of the bridge without leaving the road.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   A particle of mass $0.3$ kg is attached to one end of a light rigid rod of length $0.5$ m. The other end of the rod is freely pivoted at a fixed point $O$, and the rod can rotate in a vertical plane. The particle is projected from its lowest position with speed $u$ m s$^{-1}$ and just reaches the highest position, where it is momentarily at rest. Take $g=9.8$ m s$^{-2}$.

   (a)

   What is the value of $u$?

   [1 mark]

   • A$4.95$
   • B$2.21$
   • C$3.13$
   • D$4.43$

   (b)

   Which statement explains why the particle can reach the highest position with zero speed when it is on a rod, but would not be able to on a string?

   [1 mark]

   • AThe rod is light, so no energy is needed to lift it.
   • BThe rod can push the particle upwards at the top, balancing the weight.
   • CThe rod provides a tension towards $O$ at the top that exceeds the weight.
   • DThe rod keeps the speed of the particle constant.

   (c)

   Find the force exerted by the rod on the particle when the particle is at the highest position.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   A particle $P$ of mass $0.35$ kg is attached to one end of a light inextensible string of length $1$ m. The other end of the string is attached to a fixed point $O$. $P$ hangs at rest vertically below $O$ and is projected horizontally with speed $6$ m s$^{-1}$. Model $P$ as a particle and take $g=9.8$ m s$^{-2}$.

   (a)

   Show that the string becomes slack before $P$ reaches the highest point of the circle.

   [3 marks]

   (b)

   Find the angle above the horizontal through $O$ at which the string becomes slack.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   A smooth solid sphere of radius $0.6$ m is fixed with its lowest point on horizontal ground, and its centre is $O$. A particle $P$ of mass $0.2$ kg is placed at the highest point of the sphere and is given a negligible horizontal push, so that it begins to slide down the outer surface of the sphere. Let $\theta$ be the angle between $OP$ and the upward vertical while $P$ is on the sphere. Model $P$ as a particle and take $g=9.8$ m s$^{-2}$.

   (a)

   Show that $P$ leaves the surface of the sphere when $\cos\theta=\frac23$, and find the speed of $P$ at that instant.

   [6 marks]

   (b)

   Find the speed of $P$ when it hits the ground, and the angle its velocity makes with the horizontal at that instant.

   [6 marks]

   Total for question 8: 12 marks

End of questions