Hooke's lawEdexcel International A Level Further Maths: Subtopic test

1. 1
   A light elastic string has natural length $0.5$ m and modulus of elasticity $20$ N. One end of the string is fixed to a point $A$ and a particle $P$ of mass $0.6$ kg is attached to the other end. The particle hangs in equilibrium vertically below $A$. Take $g=9.8$ m s$^{-2}$.

   (a)

   Find the tension in the string.

   [1 mark]

   • A$0.6$ N
   • B$9.8$ N
   • C$2.94$ N
   • D$5.88$ N

   (b)

   Find the extension of the string.

   [1 mark]

   • A$0.294$ m
   • B$2.94$ m
   • C$0.147$ m
   • D$0.647$ m

   (c)

   The particle is replaced by a particle of mass $M$ kg, and in equilibrium the string has length $0.7$ m. Find $M$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A light spring has natural length $0.5$ m and modulus of elasticity $14.7$ N. The spring stands vertically with its lower end fixed to horizontal ground. A particle of mass $0.9$ kg rests in equilibrium on the top of the spring. Take $g=9.8$ m s$^{-2}$.

   (a)

   Find the magnitude of the thrust in the spring.

   [1 mark]

   • A$8.82$ N
   • B$0.9$ N
   • C$9.8$ N
   • D$14.7$ N

   (b)

   Find the length of the spring in equilibrium.

   [1 mark]

   • A$0.3$ m
   • B$0.2$ m
   • C$0.8$ m
   • D$0.5$ m

   (c)

   The particle is replaced by a particle of mass $M$ kg, and in equilibrium the length of the spring is $0.35$ m. Find $M$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Two light elastic strings $AP$ and $BP$ have natural lengths $0.6$ m and $0.5$ m and moduli of elasticity $24$ N and $30$ N respectively. The ends $A$ and $B$ are fixed to points $2$ m apart on a smooth horizontal table, and the other ends are joined to a particle $P$ lying on the table on the line $AB$, between $A$ and $B$. Both strings are taut.

   (a)

   Find the length $AP$ in equilibrium.

   [3 marks]

   (b)

   A horizontal force $F$ N acting on $P$ along $AB$ keeps $P$ in equilibrium with $AP=1.2$ m. Find the magnitude and direction of $F$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A particle $P$ of mass $3$ kg is attached to one end of a light elastic string of natural length $1.2$ m and modulus of elasticity $58.8$ N. The other end of the string is fixed to a point $O$ on a plane inclined at $30^\circ$ to the horizontal. The string lies along a line of greatest slope of the plane, with $P$ below $O$. Take $g=9.8$ m s$^{-2}$.

   (a)

   The plane is smooth, and $P$ rests in equilibrium.
   (i) Find the distance $OP$.
   (ii) The particle is replaced by one of mass $M$ kg and in equilibrium $OP=1.9$ m. Find $M$.

   [6 marks]

   (b)

   The plane is now rough, with coefficient of friction $\mu$ between $P$ and the plane. The $3$ kg particle is placed with $OP=1.8$ m and is on the point of slipping up the plane. Find the value of $\mu$.

   [6 marks]

   Total for question 4: 12 marks

End of questions