Horizontal circular motionEdexcel A-Level Further Maths: Subtopic test

1. 1
   A particle of mass 0.4 kg moves in a horizontal circle of radius 0.5 m with constant angular speed 6 rad s⁻¹.

   (a)

   Find the speed of the particle.

   [1 mark]

   • A$12$ m s⁻¹
   • B$3$ m s⁻¹
   • C$18$ m s⁻¹
   • D$0.083$ m s⁻¹

   (b)

   Find the magnitude of the acceleration of the particle.

   [1 mark]

   • A$36$ m s⁻²
   • B$9$ m s⁻²
   • C$3$ m s⁻²
   • D$18$ m s⁻²

   (c)

   Find the magnitude and direction of the resultant force on the particle.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A cyclist and bicycle, of total mass 90 kg, travel at constant speed round a circular track of radius 40 m. The track is banked at $20^\circ$ to the horizontal and the cyclist experiences no sideways frictional force. The cyclist and bicycle are modelled as a particle and $g=9.8$ m s⁻².

   (a)

   Let $R$ be the normal reaction between the track and the bicycle and $v$ the speed of the cyclist. Which pair of equations is correct?

   [1 mark]

   • A$R\cos20^\circ=90g$ and $R\sin20^\circ=\frac{90v^2}{40}$
   • B$R\sin20^\circ=90g$ and $R\cos20^\circ=\frac{90v^2}{40}$
   • C$R=90g$ and $R\sin20^\circ=\frac{90v^2}{40}$
   • D$R\cos20^\circ=90g$ and $R\sin20^\circ=\frac{90v}{40}$

   (b)

   Find the speed of the cyclist.

   [1 mark]

   • A$143$ m s⁻¹
   • B$11.6$ m s⁻¹
   • C$11.9$ m s⁻¹
   • D$32.8$ m s⁻¹

   (c)

   Find the normal reaction between the track and the bicycle.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   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 attached to a fixed point $O$. $P$ moves in a horizontal circle with constant angular speed, with the string taut and making a constant angle of $60^\circ$ with the downward vertical through $O$. Take $g=9.8$ m s⁻².

   (a)

   Find the tension in the string and the radius of the circle.

   [3 marks]

   (b)

   Find the angular speed of $P$ and the time taken for one complete revolution.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A particle $P$ of mass 0.3 kg is attached to one end of a light elastic string of natural length 0.8 m and modulus of elasticity 24 N. The other end of the string is fixed to a point $O$ on a smooth horizontal table. $P$ moves on the table in a horizontal circle with centre $O$ at constant speed, with the string taut and horizontal.

   (a)

   The radius of the circle is 1.0 m. Find (i) the tension in the string, (ii) the speed of $P$, (iii) the time taken for $P$ to complete one revolution.

   [6 marks]

   (b)

   The string will break if the tension exceeds 15 N. Find the greatest angular speed at which $P$ can move in a horizontal circle without the string breaking, and the speed of $P$ at this angular speed.

   [6 marks]

   Total for question 4: 12 marks

End of questions