Equilibrium of a particleEdexcel International A Level Maths: Subtopic test

1. 1
   A crate of mass 20 kg rests on a rough horizontal floor. A rope attached to the crate is pulled with a force of 50 N at $30^\circ$ above the horizontal, and the crate remains at rest. Take $g=9.8$ m s$^{-2}$ and model the crate as a particle.

   (a)

   Find the normal reaction of the floor on the crate.

   [1 mark]

   • A$196$ N
   • B$221$ N
   • C$171$ N
   • D$153$ N

   (b)

   Find the friction force acting on the crate.

   [1 mark]

   • A$43.3$ N
   • B$25$ N
   • C$50$ N
   • D$171$ N

   (c)

   The crate is on the point of slipping. Find the coefficient of friction between the crate and the floor.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A particle of weight 30 N is held in equilibrium on a smooth plane inclined at $30^\circ$ to the horizontal by a force of magnitude $P$ N acting up the slope, parallel to the line of greatest slope.

   (a)

   Find the value of $P$.

   [1 mark]

   • A$26.0$
   • B$15$
   • C$30$
   • D$17.3$

   (b)

   Find the normal reaction of the plane on the particle.

   [1 mark]

   • A$15$ N
   • B$30$ N
   • C$17.3$ N
   • D$26.0$ N

   (c)

   The force $P$ is removed and replaced by a horizontal force of magnitude $H$ N, so that the particle is still in equilibrium on the plane. Find $H$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A particle of weight 80 N is in equilibrium, supported by two light inextensible strings. One string has tension $T_1$ N and makes an angle of $30^\circ$ with the horizontal. The other has tension $T_2$ N and makes an angle of $60^\circ$ with the horizontal, on the opposite side of the particle.

   (a)

   By resolving horizontally, show that $T_2=\sqrt{3}\,T_1$.

   [3 marks]

   (b)

   Find the values of $T_1$ and $T_2$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A box of mass 12 kg rests on a rough horizontal floor. The coefficient of friction between the box and the floor is 0.4. Take $g=9.8$ m s$^{-2}$ and model the box as a particle.

   (a)

   A force of magnitude $P$ N pushes the box, acting at $20^\circ$ below the horizontal. The box is on the point of sliding. Find $P$.

   [6 marks]

   (b)

   The box is now pulled by a force of magnitude $Q$ N acting at $20^\circ$ above the horizontal, and the box is again on the point of sliding. Find $Q$, and explain why $Q$ is smaller than the value of $P$ found in (a).

   [6 marks]

   Total for question 4: 12 marks

End of questions