Mechanics 3 (M3): Elastic strings and springsEdexcel International A Level Further Maths: Topic test

1. 1
   A light elastic string has natural length $1.5$ m and modulus of elasticity $45$ N.

   (a)

   The string is stretched to a length of $1.8$ m. Find the tension in the string, in N.

   [1 mark]

   • A$9$
   • B$54$
   • C$13.5$
   • D$7.5$

   (b)

   The tension in the string is $24$ N. Find the extension of the string, in m to 2 significant figures.

   [1 mark]

   • A$2.3$
   • B$1.3$
   • C$0.53$
   • D$0.80$

   (c)

   A particle of mass $2$ kg hangs in equilibrium from one end of the string, the other end being fixed. Find the extension of the string, in m to 3 significant figures. Take $g=9.8$ m s$^{-2}$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A light elastic string has natural length $2$ m and modulus of elasticity $50$ N.

   (a)

   Find the elastic potential energy stored in the string when its length is $2.4$ m, in J.

   [1 mark]

   • A$4$
   • B$10$
   • C$2$
   • D$1$

   (b)

   Find the work done in stretching the string from a length of $2.4$ m to a length of $2.8$ m, in J.

   [1 mark]

   • A$8$
   • B$6$
   • C$2$
   • D$4$

   (c)

   The string lies on a smooth horizontal table with one end fixed. A particle of mass $0.5$ kg is attached to the other end and is held with the string at a length of $2.8$ m. The particle is released from rest. Find its speed when the string reaches its natural length.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A particle $P$ of mass $4$ kg hangs in equilibrium from a fixed point $O$, attached by a light elastic string of natural length $1.5$ m and modulus of elasticity $98$ N. Take $g=9.8$ m s$^{-2}$.

   (a)

   Find the extension of the string in the equilibrium position.

   [3 marks]

   (b)

   $P$ is pulled down a further $0.6$ m from the equilibrium position and released from rest. Find the speed of $P$ as it passes through the equilibrium position.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A particle $P$ of mass $2$ kg rests on a rough horizontal table. It is attached to one end of a light elastic string of natural length $0.8$ m and modulus of elasticity $40$ N. The other end of the string is fixed to a point $O$ on the table. The coefficient of friction between $P$ and the table is $0.25$. $P$ is held at rest with $OP=1.2$ m and is then released. Take $g=9.8$ m s$^{-2}$.

   (a)

   (i) Show that $P$ begins to move when it is released.
   (ii) Find the speed of $P$ when the string reaches its natural length.

   [6 marks]

   (b)

   $P$ comes to rest at the point $C$. Find the distance $OC$.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   A light spring has natural length $0.4$ m and modulus of elasticity $25$ N. Take $g=9.8$ m s$^{-2}$.

   (a)

   The spring is compressed to a length of $0.32$ m. Find the thrust in the spring, in N.

   [1 mark]

   • A$6.25$
   • B$2$
   • C$125$
   • D$5$

   (b)

   The thrust in the spring is $15$ N. Find the length of the spring, in m.

   [1 mark]

   • A$0.16$
   • B$0.24$
   • C$0.64$
   • D$0.28$

   (c)

   The spring is placed vertically on horizontal ground with a block of mass $1.5$ kg resting on its upper end in equilibrium. Find the compression of the spring, in m to 3 significant figures.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   A particle $P$ of mass $1.5$ kg is attached to one end of a light elastic string of natural length $1.2$ m and modulus of elasticity $36$ N. The other end of the string is fixed to a point $O$ on a horizontal table. $P$ is held at rest on the table with the string stretched to a length of $1.7$ m and is then released. Take $g=9.8$ m s$^{-2}$.

   (a)

   Find the elastic potential energy stored in the string when $P$ is released, in J.

   [1 mark]

   • A$7.5$
   • B$15$
   • C$3.75$
   • D$1.875$

   (b)

   The table is smooth. Find the speed of $P$ when the string reaches its natural length, in m s$^{-1}$ to 3 significant figures.

   [1 mark]

   • A$5.00$
   • B$2.24$
   • C$3.16$
   • D$1.58$

   (c)

   The table is now rough and the coefficient of friction between $P$ and the table is $0.2$. Find the speed of $P$ when the string reaches its natural length, in m s$^{-1}$ to 3 significant figures.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   A particle $P$ of mass $3$ kg lies on a smooth plane inclined at $30^\circ$ to the horizontal. $P$ is attached to one end of a light elastic string of natural length $0.8$ m and modulus of elasticity $49$ N. The other end of the string is fixed to a point $A$ at the top of the plane, and the string lies along a line of greatest slope. $P$ rests in equilibrium below $A$. Take $g=9.8$ m s$^{-2}$.

   (a)

   Find the extension of the string.

   [3 marks]

   (b)

   $P$ is pulled a further $0.36$ m down the plane and released from rest. Find the speed of $P$ when the string first reaches its natural length.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   A particle $P$ of mass $0.5$ kg is attached to one end of a light elastic string of natural length $0.8$ m and modulus of elasticity $19.6$ N. The other end of the string is attached to a fixed point $O$. $P$ is released from rest at $O$ and falls vertically. Take $g=9.8$ m s$^{-2}$.

   (a)

   (i) Show that the greatest extension of the string is $0.8$ m.
   (ii) Find the speed of $P$ when the extension of the string is $0.4$ m.

   [6 marks]

   (b)

   (i) Show that the speed of $P$ is greatest when the extension of the string is $0.2$ m.
   (ii) Find the greatest speed of $P$.

   [6 marks]

   Total for question 8: 12 marks

End of questions