Mechanics: Forces and Newton's lawsEdexcel A-Level Maths: Topic test

1. 1
   A crane lifts a steel beam of mass $600$ kg using a single vertical cable. The beam is modelled as a particle and $g=9.8$ m s$^{-2}$. Air resistance is negligible.

   (a)

   The beam is raised at a constant velocity of $0.5$ m s$^{-1}$. What is the tension in the cable?

   [1 mark]

   • A$0$ N
   • B$5880$ N
   • C$6180$ N
   • D$5580$ N

   (b)

   While the beam is rising at $0.5$ m s$^{-1}$ the cable snaps. What is the acceleration of the beam immediately afterwards?

   [1 mark]

   • A$0$, because it was moving at constant velocity
   • B$9.8$ m s$^{-2}$ upwards
   • C$9.8$ m s$^{-2}$ downwards
   • D$0.5$ m s$^{-2}$ downwards

   (c)

   The beam is now raised with an upward acceleration of $0.25$ m s$^{-2}$. Find the tension in the cable.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A pebble is dropped from rest at the top of a vertical well and hits the water $2.5$ s later. Model the pebble as a particle moving freely under gravity, with $g=9.8$ m s$^{-2}$.

   (a)

   What is the speed of the pebble as it hits the water?

   [1 mark]

   • A$9.8$ m s$^{-1}$
   • B$61.3$ m s$^{-1}$
   • C$12.3$ m s$^{-1}$
   • D$24.5$ m s$^{-1}$

   (b)

   What is the depth of the water surface below the top of the well?

   [1 mark]

   • A$30.6$ m
   • B$61.3$ m
   • C$24.5$ m
   • D$15.3$ m

   (c)

   Find the time taken for the pebble to fall the first $10$ m.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A car of mass $1000$ kg tows a caravan of mass $600$ kg along a straight horizontal road, using a light rigid tow-bar. The driving force of the car's engine is $2600$ N. The resistance to motion is a constant $400$ N on the car and a constant $200$ N on the caravan.

   (a)

   Find the acceleration of the car and caravan.

   [3 marks]

   (b)

   Find the tension in the tow-bar, and explain why the tow-bar pulls on the car with a force of the same magnitude as it pulls on the caravan.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A block $P$ of mass $5$ kg rests on a rough plane inclined at an angle $\alpha$ to the horizontal, where $\tan\alpha=\frac34$. The block is on the point of sliding down the plane. Take $g=9.8$ m s$^{-2}$.

   (a)

   (i) Find the normal reaction between $P$ and the plane. (ii) Find the coefficient of friction between $P$ and the plane. (iii) State, with a reason, what would happen if the mass of $P$ were doubled and it were placed at rest on the same plane.

   [6 marks]

   (b)

   A horizontal force of magnitude $X$ newtons, acting in the vertical plane containing a line of greatest slope and directed towards the plane, is now applied to $P$. The block is now on the point of sliding up the plane. Find $X$.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   A particle of mass $3$ kg is held at rest on a smooth plane inclined at $40^\circ$ to the horizontal by a horizontal force of magnitude $H$ newtons. The force acts in the vertical plane containing a line of greatest slope and is directed towards the plane. Take $g=9.8$ m s$^{-2}$.

   (a)

   What is the value of $H$?

   [1 mark]

   • A$18.9$ N
   • B$22.5$ N
   • C$24.7$ N
   • D$35.0$ N

   (b)

   What is the magnitude of the normal reaction between the particle and the plane?

   [1 mark]

   • A$38.4$ N
   • B$29.4$ N
   • C$22.5$ N
   • D$18.9$ N

   (c)

   The horizontal force is removed. Find the acceleration of the particle.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   Two horizontal forces, of magnitudes $7$ N and $5$ N, act on a particle of mass $2$ kg. The angle between the lines of action of the two forces is $60^\circ$. No other horizontal forces act on the particle.

   (a)

   What is the magnitude of the resultant of the two forces?

   [1 mark]

   • A$12.0$ N
   • B$6.24$ N
   • C$8.60$ N
   • D$10.4$ N

   (b)

   What is the angle between the resultant and the $7$ N force?

   [1 mark]

   • A$35.5^\circ$
   • B$24.5^\circ$
   • C$30^\circ$
   • D$31.7^\circ$

   (c)

   Find the magnitude of the acceleration of the particle.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   A student pushes a loaded trolley of mass $12$ kg along rough horizontal ground by applying a force of $60$ N along the handle, which is inclined at $25^\circ$ below the horizontal. The coefficient of friction between the trolley and the ground is $0.25$. Model the trolley as a particle and take $g=9.8$ m s$^{-2}$.

   (a)

   Find the normal reaction between the trolley and the ground.

   [3 marks]

   (b)

   Find the acceleration of the trolley.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   Particle $A$ of mass $4$ kg rests on a rough horizontal table, the coefficient of friction between $A$ and the table being $0.5$. A light inextensible string attached to $A$ passes over a smooth light pulley at the edge of the table and is attached to a particle $B$ of mass $6$ kg, which hangs freely $1.5$ m above the floor. The string is taut and the system is released from rest. Take $g=9.8$ m s$^{-2}$.

   (a)

   Find the acceleration of the particles and the tension in the string while $B$ is falling.

   [6 marks]

   (b)

   $B$ hits the floor without rebounding. $A$ does not reach the pulley. Find the further distance that $A$ moves before it comes to rest.

   [6 marks]

   Total for question 8: 12 marks

End of questions