Work and power with forces at an angleAQA A-Level Further Maths: Subtopic test

1. 1
   A child pulls a sledge of mass 15 kg in a straight line along horizontal ground by a rope inclined at $25^\circ$ above the horizontal. The tension in the rope is 40 N and the sledge moves 30 m. A constant horizontal resistance of 28 N acts on the sledge. The sledge starts from rest.

   (a)

   Find the work done by the tension in the rope.

   [1 mark]

   • A$507$ J
   • B$1200$ J
   • C$1320$ J
   • D$1090$ J

   (b)

   Find the speed of the sledge after it has moved 30 m.

   [1 mark]

   • A$6.93\text{ m s}^{-1}$
   • B$5.75\text{ m s}^{-1}$
   • C$12.0\text{ m s}^{-1}$
   • D$33.0\text{ m s}^{-1}$

   (c)

   Find the power developed by the rope at the instant when the sledge is moving at $5\text{ m s}^{-1}$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A car of mass 1000 kg travels up a straight road inclined at an angle $\theta$ to the horizontal, where $\sin\theta=0.1$. The resistance to motion is a constant 400 N and the engine works at a constant 25 kW. Take $g=9.8\text{ m s}^{-2}$.

   (a)

   Find the greatest steady speed at which the car can climb the road.

   [1 mark]

   • A$18.1\text{ m s}^{-1}$
   • B$62.5\text{ m s}^{-1}$
   • C$25.5\text{ m s}^{-1}$
   • D$0.0552\text{ m s}^{-1}$

   (b)

   Find the acceleration of the car when its speed is $12\text{ m s}^{-1}$.

   [1 mark]

   • A$1.68\text{ m s}^{-2}$
   • B$2.08\text{ m s}^{-2}$
   • C$0.703\text{ m s}^{-2}$
   • D$1.10\text{ m s}^{-2}$

   (c)

   Find the power the engine must develop for the car to climb at a constant $15\text{ m s}^{-1}$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A box of mass 20 kg is pulled from rest up a line of greatest slope of a smooth ramp inclined at $20^\circ$ to the horizontal. The pulling force is a constant 120 N, applied by a rope that makes an angle of $30^\circ$ with the ramp, above it. The box moves 6 m up the ramp. Take $g=9.8\text{ m s}^{-2}$.

   (a)

   Find the work done by the rope and the gain in gravitational potential energy of the box.

   [3 marks]

   (b)

   Find the speed of the box after it has moved 6 m up the ramp.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A lorry of mass 5000 kg climbs a straight hill inclined at an angle $\alpha$ to the horizontal, where $\sin\alpha=\frac{1}{25}$. The resistance to motion is a constant 1500 N and the engine has a maximum power of 80 kW. Take $g=9.8\text{ m s}^{-2}$.

   (a)

   (i) Find the greatest steady speed of the lorry up the hill when the engine works at 80 kW.
   (ii) Find the acceleration of the lorry up the hill at the instant its speed is $15\text{ m s}^{-1}$ with the engine at 80 kW.

   [6 marks]

   (b)

   The driver now uses less than full power so that the lorry climbs the hill at a constant $20\text{ m s}^{-1}$ for 3 minutes. Find the work done by the engine in this time, and show that it equals the gain in potential energy plus the work done against the resistance.

   [6 marks]

   Total for question 4: 12 marks

End of questions