Forces as vectors and resolving forcesEdexcel International A Level Maths: Subtopic test

1. 1
   A force $\mathbf{F}$ of magnitude 20 N acts in the $x$–$y$ plane at $60^\circ$ above the positive $x$-axis. The unit vectors $\mathbf{i}$ and $\mathbf{j}$ are in the directions of the positive $x$-axis and $y$-axis.

   (a)

   Find the component of $\mathbf{F}$ in the $x$-direction.

   [1 mark]

   • A$17.3$ N
   • B$10$ N
   • C$20$ N
   • D$40$ N

   (b)

   Find the component of $\mathbf{F}$ in the $y$-direction.

   [1 mark]

   • A$10$ N
   • B$20$ N
   • C$17.3$ N
   • D$23.1$ N

   (c)

   Write $\mathbf{F}$ in the form $a\mathbf{i}+b\mathbf{j}$, giving $a$ and $b$ exactly.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Two forces act on a particle: $\mathbf{F}_1=(3\mathbf{i}+4\mathbf{j})$ N and $\mathbf{F}_2=(-7\mathbf{i}+2\mathbf{j})$ N.

   (a)

   Find the resultant of the two forces.

   [1 mark]

   • A$(10\mathbf{i}+2\mathbf{j})$ N
   • B$(-4\mathbf{i}+2\mathbf{j})$ N
   • C$(4\mathbf{i}-6\mathbf{j})$ N
   • D$(-4\mathbf{i}+6\mathbf{j})$ N

   (b)

   Find the magnitude of the resultant of the two forces.

   [1 mark]

   • A$\sqrt{52}$ N
   • B$10$ N
   • C$52$ N
   • D$2$ N

   (c)

   Find the angle between the resultant and the vector $\mathbf{j}$, giving your answer to 1 decimal place.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Three forces act on a particle: $\mathbf{F}_1=(2\mathbf{i}-3\mathbf{j})$ N, $\mathbf{F}_2=(p\mathbf{i}+q\mathbf{j})$ N and $\mathbf{F}_3=(-\mathbf{i}+5\mathbf{j})$ N, where $p$ and $q$ are constants. The resultant of the three forces is $(6\mathbf{i}+4\mathbf{j})$ N.

   (a)

   Find the values of $p$ and $q$.

   [3 marks]

   (b)

   Find the magnitude of $\mathbf{F}_2$ and the angle that $\mathbf{F}_2$ makes with the vector $\mathbf{i}$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Horizontal forces act on a particle at $O$. The unit vectors $\mathbf{i}$ and $\mathbf{j}$ point due east and due north. A force $\mathbf{P}$ of magnitude 10 N acts $30^\circ$ north of east. A force $\mathbf{Q}$ of magnitude 6 N acts $60^\circ$ north of west.

   (a)

   (i) Write $\mathbf{P}$ and $\mathbf{Q}$ in the form $a\mathbf{i}+b\mathbf{j}$, giving exact values.
   (ii) Find the magnitude of the resultant of $\mathbf{P}$ and $\mathbf{Q}$, and the angle it makes with due east.

   [6 marks]

   (b)

   A third force $\mathbf{T}$ is applied at $O$ so that the resultant of $\mathbf{P}$, $\mathbf{Q}$ and $\mathbf{T}$ is a force of magnitude 3 N acting due east. Find the magnitude of $\mathbf{T}$ and its direction, giving the angle west of due south.

   [6 marks]

   Total for question 4: 12 marks

End of questions