Resolving forces and resultantsAQA A-Level Maths: Subtopic test

1. 1
   Three coplanar forces $\mathbf{P}=(4\mathbf{i}-3\mathbf{j})$ N, $\mathbf{Q}=(-7\mathbf{i}+5\mathbf{j})$ N and $\mathbf{R}$ act on a particle, which is in equilibrium. The vectors $\mathbf{i}$ and $\mathbf{j}$ are perpendicular unit vectors.

   (a)

   Find $\mathbf{R}$.

   [1 mark]

   • A$(-3\mathbf{i}+2\mathbf{j})$ N
   • B$(11\mathbf{i}-8\mathbf{j})$ N
   • C$(3\mathbf{i}-2\mathbf{j})$ N
   • D$(3\mathbf{i}+2\mathbf{j})$ N

   (b)

   Find the magnitude of $\mathbf{R}$.

   [1 mark]

   • A$\sqrt{13}$ N
   • B$13$ N
   • C$5$ N
   • D$1$ N

   (c)

   Find the angle that $\mathbf{R}$ makes with the direction of $\mathbf{i}$, and state whether it is above or below that direction.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A block of mass $6$ kg is released from rest on a smooth plane inclined at $30^\circ$ to the horizontal. Model the block as a particle, with $g=9.8\ \text{m s}^{-2}$.

   (a)

   Find the component of the block's weight acting down the plane.

   [1 mark]

   • A$50.9$ N
   • B$58.8$ N
   • C$4.9$ N
   • D$29.4$ N

   (b)

   Find the normal reaction of the plane on the block.

   [1 mark]

   • A$29.4$ N
   • B$50.9$ N
   • C$58.8$ N
   • D$8.49$ N

   (c)

   Find the acceleration of the block down the plane.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Two horizontal forces $(6\mathbf{i}+4\mathbf{j})$ N and $(-2\mathbf{i}+8\mathbf{j})$ N act on a particle of mass $2$ kg on a smooth horizontal surface, where $\mathbf{i}$ and $\mathbf{j}$ are perpendicular horizontal unit vectors. These are the only horizontal forces. The particle starts from rest at the origin $O$.

   (a)

   Find the acceleration of the particle as a vector, and its magnitude.

   [3 marks]

   (b)

   Find the position vector of the particle after $3$ s, and its distance from $O$ at that time.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A lamp of mass $12$ kg hangs in equilibrium from a horizontal ceiling by two light inextensible strings. One string makes an angle of $30^\circ$ with the ceiling and the other makes an angle of $60^\circ$ with the ceiling, on the opposite side of the lamp. Take $g=9.8\ \text{m s}^{-2}$.

   (a)

   Find the tension in each string.

   [6 marks]

   (b)

   The strings are replaced by two identical strings that make equal angles $\theta$ with the ceiling, one on each side of the lamp. Each string can withstand a tension of at most $100$ N. Find the smallest possible value of $\theta$.

   [6 marks]

   Total for question 4: 12 marks

End of questions