Further Mechanics 1: Work, energy and powerAQA A-Level Further Maths: Topic test

1. 1
   A parcel of mass $6$ kg is released from rest at the top of a straight rough chute. The chute is $10$ m long and the top is $4$ m higher than the bottom. The parcel reaches the bottom with a speed of $5$ m s⁻¹. Take $g=9.8$ m s⁻².

   (a)

   Find the loss in gravitational potential energy of the parcel between the top and the bottom of the chute, in J.

   [1 mark]

   • A$24$
   • B$235.2$
   • C$588$
   • D$75$

   (b)

   Find the work done against friction as the parcel slides down the chute, in J.

   [1 mark]

   • A$75$
   • B$235.2$
   • C$310.2$
   • D$160.2$

   (c)

   Find the magnitude of the frictional force on the parcel, which may be assumed to be constant.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A light spring of natural length $0.50$ m and stiffness $240$ N m⁻¹ is compressed to a length of $0.30$ m.

   (a)

   Find the force needed to hold the spring at this length, in N.

   [1 mark]

   • A$48$
   • B$120$
   • C$72$
   • D$24$

   (b)

   Find the elastic potential energy stored in the spring, in J.

   [1 mark]

   • A$9.6$
   • B$24$
   • C$4.8$
   • D$10.8$

   (c)

   A block of mass $0.6$ kg rests against the compressed spring on a smooth horizontal surface, and the spring is released. Find the speed of the block when it leaves the spring, which has returned to its natural length.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A crate of mass $3$ kg is at rest at the origin on a smooth horizontal floor. It is pulled along a straight line by a horizontal force of magnitude $(16-x^{2})$ newtons, where $x$ metres is the displacement of the crate from the origin.

   (a)

   Find the work done by the force as the crate moves from $x=0$ to $x=3$.

   [3 marks]

   (b)

   Find the speed of the crate when $x=3$ and the power developed by the force at that instant. Give each answer to 3 significant figures.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A kite buggy of total mass $90$ kg is initially at rest on level sand. The kite line exerts a force of $300$ N on the buggy, inclined at $35^{\circ}$ above the horizontal in the direction of motion. The total resistance to the motion of the buggy is a constant $140$ N.

   (a)

   Find the work done by the kite line as the buggy moves $50$ m in a straight line, and hence find the speed of the buggy after $50$ m. Give the speed to 3 significant figures.

   [6 marks]

   (b)

   Find the power developed by the kite line at the instant when the buggy has covered $50$ m. Later the wind drops and the buggy moves at a constant speed of $8$ m s⁻¹ with the line still inclined at $35^{\circ}$ above the horizontal. Find the tension in the line and the power it then develops. Give the powers to 3 significant figures.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   A lift of mass $600$ kg is raised by a vertical cable driven by a motor. Take $g=9.8$ m s⁻².

   (a)

   The lift rises at a constant speed of $2$ m s⁻¹. Find the tension in the cable, in N.

   [1 mark]

   • A$600$
   • B$1200$
   • C$11\,760$
   • D$5880$

   (b)

   Find the power developed by the motor when the lift rises at a constant speed of $2$ m s⁻¹, in W.

   [1 mark]

   • A$5880$
   • B$11\,760$
   • C$2940$
   • D$1200$

   (c)

   The lift now accelerates upwards at $0.5$ m s⁻². Find the power developed by the motor at the instant when the speed is $2$ m s⁻¹. Give your answer to 3 significant figures.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   A bungee jumper of mass $70$ kg is attached to one end of a light elastic rope of natural length $20$ m and modulus of elasticity $8232$ N. The other end of the rope is fixed to a platform. The jumper steps off the platform and falls vertically from rest. Treat the jumper as a particle, ignore air resistance, and take $g=9.8$ m s⁻².

   (a)

   Find the speed of the jumper at the instant the rope becomes taut, in m s⁻¹.

   [1 mark]

   • A$392$
   • B$14.0$
   • C$19.8$
   • D$28.0$

   (b)

   At the lowest point of the fall the extension of the rope is $10$ m. Find the elastic potential energy stored in the rope at this point, in J.

   [1 mark]

   • A$20\,580$
   • B$2058$
   • C$41\,160$
   • D$411\,600$

   (c)

   Find the magnitude of the acceleration of the jumper at the lowest point, where the extension of the rope is $10$ m.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   A motorcycle and rider have a total mass of $250$ kg. The motorcycle travels along a straight horizontal road with its engine working at a constant rate of $20$ kW. The resistance to motion is a constant $400$ N.

   (a)

   Find the driving force of the engine and the acceleration of the motorcycle at the instant when its speed is $25$ m s⁻¹.

   [3 marks]

   (b)

   Find the maximum speed of the motorcycle on this road. Find the work done by the engine in $30$ seconds at this maximum speed.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   A block of mass $2$ kg lies on a rough horizontal table. The coefficient of friction between the block and the table is $0.25$. The block is attached to one end of a light elastic string of natural length $0.5$ m and modulus of elasticity $40$ N. The other end of the string is fixed to a point $O$ on the table. The block is held at rest on the table at a distance of $0.8$ m from $O$, with the string straight, and is then released. Take $g=9.8$ m s⁻².

   (a)

   Find the speed of the block at the instant when the string becomes slack. Give your answer to 3 significant figures.

   [6 marks]

   (b)

   After the string becomes slack the block continues to slide until it comes to rest. Find the distance it slides after the string becomes slack. Find also the total work done against friction from release until the block stops, and explain how this compares with the initial elastic potential energy.

   [6 marks]

   Total for question 8: 12 marks

End of questions