Further Mechanics 1: Circular motionAQA A-Level Further Maths: Topic test

1. 1
   A laboratory centrifuge rotates at a constant $3000$ revolutions per minute. A sample tube is held with its centre $0.12$ m from the axis of rotation. The tube is modelled as a particle.

   (a)

   What is the angular speed of the centrifuge?

   [1 mark]

   • A$100\pi$ rad s⁻¹
   • B$50$ rad s⁻¹
   • C$6000\pi$ rad s⁻¹
   • D$50\pi$ rad s⁻¹

   (b)

   What is the speed of the tube?

   [1 mark]

   • A$18.8$ m s⁻¹
   • B$37.7$ m s⁻¹
   • C$2620$ m s⁻¹
   • D$377$ m s⁻¹

   (c)

   Find the magnitude of the acceleration of the tube.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A particle $P$ moves on a circle with centre the origin $O$ and radius $5$ m. At time $t$ seconds its position vector relative to $O$ is $\mathbf{r}=5\cos(0.4t)\,\mathbf{i}+5\sin(0.4t)\,\mathbf{j}$, where the unit vectors $\mathbf{i}$ and $\mathbf{j}$ are in the plane of the circle.

   (a)

   What is the speed of $P$?

   [1 mark]

   • A$0.4$ m s⁻¹
   • B$5$ m s⁻¹
   • C$2$ m s⁻¹
   • D$12.5$ m s⁻¹

   (b)

   Which statement about the velocity $\mathbf{v}$ and the position vector $\mathbf{r}$ of $P$ is true for all values of $t$?

   [1 mark]

   • A$\mathbf{v}$ is parallel to $\mathbf{r}$ and in the same direction
   • B$\mathbf{v}$ is parallel to $\mathbf{r}$ and in the opposite direction
   • C$\mathbf{v}$ is directed towards $O$
   • D$\mathbf{v}$ is perpendicular to $\mathbf{r}$

   (c)

   Show that the acceleration of $P$ is $-0.16\,\mathbf{r}$ and state its magnitude.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A ball 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 fixed at a point $O$. The ball moves in a horizontal circle with constant angular speed $4$ rad s⁻¹, with the string taut and inclined at an angle $\theta$ to the downward vertical through $O$.

   (a)

   Show that the tension in the string is $3.6$ N.

   [3 marks]

   (b)

   Find the angle $\theta$ and the radius of the circle in which the ball moves.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A particle $P$ of mass $0.5$ kg is attached to one end of a light inextensible string of length $1.2$ m. The other end of the string is fixed at a point $O$. The particle hangs at rest vertically below $O$ and is then projected horizontally with speed $u$ m s⁻¹ so that it moves in a vertical circle with centre $O$.

   (a)

   Find the least value of $u$ for which the string remains taut throughout the motion.

   [6 marks]

   (b)

   The particle is instead projected with speed $9$ m s⁻¹. Find the tension in the string (i) when $P$ is at the highest point, (ii) when $OP$ is horizontal.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   A roller-coaster car of mass $350$ kg runs on the inside of a smooth vertical circular loop of radius $9$ m. The car is modelled as a particle.

   (a)

   What is the least speed the car can have at the highest point of the loop and still stay in contact with the track?

   [1 mark]

   • A$88.2$ m s⁻¹
   • B$13.3$ m s⁻¹
   • C$6.64$ m s⁻¹
   • D$9.39$ m s⁻¹

   (b)

   What is the least speed the car can have at the lowest point of the loop for it to complete the circle?

   [1 mark]

   • A$21.0$ m s⁻¹
   • B$9.39$ m s⁻¹
   • C$16.3$ m s⁻¹
   • D$18.8$ m s⁻¹

   (c)

   The car passes the highest point with speed $12$ m s⁻¹. Find the magnitude of the normal reaction of the track on the car at this point.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   A small bell of mass $0.08$ kg hangs from a fixed point $O$ on a ceiling by a light inextensible string of length $0.5$ m. The bell moves in a horizontal circle at constant speed, with the string making a constant angle of $30^{\circ}$ with the downward vertical.

   (a)

   What is the tension in the string?

   [1 mark]

   • A$0.784$ N
   • B$0.453$ N
   • C$0.905$ N
   • D$1.57$ N

   (b)

   What is the speed of the bell?

   [1 mark]

   • A$1.41$ m s⁻¹
   • B$1.68$ m s⁻¹
   • C$2.06$ m s⁻¹
   • D$1.19$ m s⁻¹

   (c)

   Find the angular speed of the bell in rad s⁻¹.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   A particle $P$ moves anticlockwise with constant speed $6$ m s⁻¹ on a circle of radius $3$ m with centre at the origin $O$. At time $t=0$ the particle is at the point with position vector $3\mathbf{i}$, where $\mathbf{i}$ and $\mathbf{j}$ are perpendicular unit vectors in the plane of the circle.

   (a)

   Find the angular speed of $P$ and an expression for the position vector of $P$ at time $t$ seconds.

   [3 marks]

   (b)

   Find the acceleration of $P$ when $t=\dfrac{\pi}{4}$, giving its magnitude and its direction.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   A particle $P$ of mass $0.35$ kg is attached to one end of a light inextensible string of length $1.4$ m. The other end of the string is fixed at a point $O$. The particle moves in a horizontal circle at constant angular speed, with the string taut and inclined at an angle $\theta$ to the downward vertical through $O$.

   (a)

   The tension in the string is $6$ N. Find $\theta$ and the angular speed of $P$ in rad s⁻¹.

   [6 marks]

   (b)

   The particle is made to rotate faster until the tension is $20$ N. Find the new value of $\theta$ and the angular speed in revolutions per minute.

   [6 marks]

   Total for question 8: 12 marks

End of questions