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

1. 1
   A cyclist and bicycle, of total mass $80$ kg, ride at a constant speed of $5$ m s$^{-1}$ up a straight road inclined at an angle $\theta$ to the horizontal, where $\sin\theta=\frac{1}{20}$. The total resistance to motion is constant at $30$ N. Take $g=9.8$ m s$^{-2}$.

   (a)

   What is the magnitude of the driving force the cyclist must provide?

   [1 mark]

   • A$39.2$ N
   • B$9.2$ N
   • C$69.2$ N
   • D$814$ N

   (b)

   What is the power developed by the cyclist?

   [1 mark]

   • A$346$ W
   • B$13.8$ W
   • C$1730$ W
   • D$196$ W

   (c)

   Find the work done against gravity as the cyclist travels $200$ m up the road.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A sledge of mass $10$ kg slides from rest down a line of greatest slope of a rough plane, inclined at $30^\circ$ to the horizontal. The sledge travels $6$ m down the plane and has speed $6$ m s$^{-1}$ at the bottom. The sledge is modelled as a particle. Take $g=9.8$ m s$^{-2}$.

   (a)

   What is the kinetic energy of the sledge at the bottom of the plane?

   [1 mark]

   • A$60$ J
   • B$180$ J
   • C$360$ J
   • D$36$ J

   (b)

   What is the loss in gravitational potential energy of the sledge as it moves down the plane?

   [1 mark]

   • A$588$ J
   • B$509$ J
   • C$180$ J
   • D$294$ J

   (c)

   Find the magnitude of the constant resistance to the motion of the sledge.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A tram of mass $12\,000$ kg is driven along a straight horizontal track by an engine working at a constant maximum power of $60$ kW. The resistance to the tram's motion is constant at $3000$ N.

   (a)

   Find the maximum speed of the tram on the horizontal track.

   [3 marks]

   (b)

   Find the acceleration of the tram when it is travelling at $10$ m s$^{-1}$ with the engine working at maximum power.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A van of mass $2000$ kg has an engine that works at a constant power of $40$ kW. The van travels up a straight hill inclined at an angle $\alpha$ to the horizontal, where $\sin\alpha=\frac{1}{20}$. The resistance to motion is constant at $1020$ N. Take $g=9.8$ m s$^{-2}$.

   (a)

   (i) Find the greatest steady speed of the van up the hill. (ii) Find the acceleration of the van when it is travelling up the hill at $16$ m s$^{-1}$.

   [6 marks]

   (b)

   The van is travelling up the hill at $20$ m s$^{-1}$ when the engine is switched off. Use the work-energy principle to find the distance the van travels up the hill before it comes to rest, and find the work done against resistance and the gain in gravitational potential energy over this distance.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   A crane lifts a concrete beam of mass $500$ kg from rest on the ground to a height of $12$ m, where the beam is moving upwards at $2$ m s$^{-1}$. The beam is modelled as a particle. Take $g=9.8$ m s$^{-2}$.

   (a)

   What is the gain in gravitational potential energy of the beam?

   [1 mark]

   • A$6000$ J
   • B$4900$ J
   • C$59\,800$ J
   • D$58\,800$ J

   (b)

   What is the gain in kinetic energy of the beam?

   [1 mark]

   • A$2000$ J
   • B$1000$ J
   • C$500$ J
   • D$4000$ J

   (c)

   Find the work done by the crane in lifting the beam.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   A pump raises water from a well and discharges it at ground level, $15$ m above the surface of the water. The pump raises $20$ kg of water each second, and the water leaves the pump with speed $4$ m s$^{-1}$. The water leaves the surface of the well with negligible speed. Take $g=9.8$ m s$^{-2}$.

   (a)

   What is the gain in gravitational potential energy of the water each second?

   [1 mark]

   • A$2940$ J
   • B$294$ J
   • C$196$ J
   • D$300$ J

   (b)

   What is the power the pump must supply to the water?

   [1 mark]

   • A$160$ W
   • B$2940$ W
   • C$3100$ W
   • D$3260$ W

   (c)

   Find the work done by the pump on the water in $2$ minutes.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   A box of mass $6$ kg is projected up a line of greatest slope of a rough plane at $7$ m s$^{-1}$. The plane is inclined at an angle $\alpha$ to the horizontal, where $\tan\alpha=\frac34$, and the coefficient of friction between the box and the plane is $0.25$. The box is modelled as a particle. Take $g=9.8$ m s$^{-2}$.

   (a)

   Find the magnitude of the friction force acting on the box.

   [3 marks]

   (b)

   Use the work-energy principle to find the distance the box travels up the plane before it first comes to rest.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   A skier of mass $65$ kg is modelled as a particle in each part of this question. Take $g=9.8$ m s$^{-2}$.

   (a)

   The skier starts from rest at a point $A$ on a ski slope and travels to a point $B$ that is $30$ m vertically below $A$. The speed of the skier at $B$ is $20$ m s$^{-1}$. Find the work done against resistance between $A$ and $B$.

   [6 marks]

   (b)

   The skier is then pulled at a constant speed of $2$ m s$^{-1}$ up a straight slope inclined at an angle $\beta$ to the horizontal, where $\sin\beta=0.2$, by a tow rope parallel to the slope. The resistance to motion is $40$ N. Find the power supplied through the rope, and the work done by the rope as the skier travels $150$ m up the slope.

   [6 marks]

   Total for question 8: 12 marks

End of questions