Elastic potential energyEdexcel International A Level Further Maths: Subtopic test

1. 1
   A light elastic string has natural length $0.6$ m and modulus of elasticity $30$ N. The string is stretched to a length of $0.9$ m.

   (a)

   Find the tension in the string.

   [1 mark]

   • A$10$ N
   • B$15$ N
   • C$30$ N
   • D$45$ N

   (b)

   Find the elastic potential energy stored in the string.

   [1 mark]

   • A$4.5$ J
   • B$13.5$ J
   • C$20.25$ J
   • D$2.25$ J

   (c)

   The string is now stretched slowly from a length of $0.9$ m to a length of $1.1$ m. Find the work done against the tension of the string.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A particle $P$ of mass $0.5$ kg is attached to one end of a light elastic spring of natural length $0.4$ m and modulus of elasticity $20$ N. The other end of the spring is fixed to a point $O$ on a smooth horizontal table. $P$ is held on the table at a distance $0.6$ m from $O$ and released from rest.

   (a)

   Find the elastic potential energy stored in the spring when $P$ is released.

   [1 mark]

   • A$1$ J
   • B$2$ J
   • C$9$ J
   • D$4$ J

   (b)

   Find the speed of $P$ when the spring reaches its natural length.

   [1 mark]

   • A$4$ m s$^{-1}$
   • B$\sqrt2$ m s$^{-1}$
   • C$2$ m s$^{-1}$
   • D$2\sqrt2$ m s$^{-1}$

   (c)

   Find the speed of $P$ when it is $0.5$ m from $O$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A particle $P$ of mass $2$ kg is attached to one end of a light elastic string of natural length $1.5$ m and modulus of elasticity $60$ N. The other end of the string is fixed to a point $A$ on a ceiling. $P$ is released from rest at $A$ and falls vertically. Take $g=9.8$ m s$^{-2}$ and ignore air resistance.

   (a)

   Show that, when $P$ first comes to instantaneous rest, the extension $x$ metres of the string satisfies $20x^2-19.6x-29.4=0$.

   [3 marks]

   (b)

   Find the speed of $P$ when it is $2$ m below $A$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A particle $P$ of mass $0.4$ kg is attached to one end of a light elastic string of natural length $0.5$ m and modulus of elasticity $15$ N. The other end of the string is fixed to a point $O$ on a smooth plane inclined at $30^\circ$ to the horizontal, with $O$ at the top of the plane. $P$ is released from rest at the point $A$ on the plane where $OA=0.5$ m, and slides down the line of greatest slope through $O$. Take $g=9.8$ m s$^{-2}$.

   (a)

   (i) Find the speed of $P$ when the extension of the string is $0.1$ m.
   (ii) Find the greatest extension of the string.

   [6 marks]

   (b)

   Find the maximum speed of $P$. The string will break if the tension exceeds $12$ N. Determine whether the string breaks.

   [6 marks]

   Total for question 4: 12 marks

End of questions