M1: Statics of a particleEdexcel International A Level Maths: Topic test

1. 1
   A force of magnitude $26$ N acts on a particle in the vertical plane. The force makes an angle $\theta$ with the horizontal, where $\tan\theta=\frac{5}{12}$ and the force has a horizontal component directed to the right and a vertical component directed upwards. A second force of magnitude $7$ N acts on the particle horizontally to the left.

   (a)

   Find the horizontal component of the $26$ N force.

   [1 mark]

   • A$10$ N
   • B$24$ N
   • C$10.8$ N
   • D$62.4$ N

   (b)

   Find the magnitude of the resultant of the two forces.

   [1 mark]

   • A$19$ N
   • B$27$ N
   • C$32.6$ N
   • D$19.7$ N

   (c)

   Find the angle the resultant makes with the horizontal.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A particle of weight $45$ N is held in equilibrium by two light strings. One string is horizontal with tension $X$ N. The other string is inclined at an angle $\alpha$ to the horizontal, where $\sin\alpha=\frac35$, and has tension $T$ N.

   (a)

   Find the value of $T$.

   [1 mark]

   • A$75$
   • B$27$
   • C$56.3$
   • D$36$

   (b)

   Find the value of $X$.

   [1 mark]

   • A$33.8$
   • B$75$
   • C$60$
   • D$36$

   (c)

   The inclined string breaks if its tension exceeds $100$ N. Find the greatest weight the particle can have for the strings to remain in this arrangement without the inclined string breaking.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A box of mass $8$ kg rests in equilibrium on a rough plane inclined at $25^\circ$ to the horizontal. The coefficient of friction between the box and the plane is $0.5$. Take $g=9.8$ m s⁻².

   (a)

   Find the magnitudes of the frictional force and the normal reaction acting on the box.

   [3 marks]

   (b)

   A horizontal force of magnitude $X$ N, in the vertical plane containing a line of greatest slope, is applied to the box, directed towards the plane. The box is on the point of sliding up the plane. Find the value of $X$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A particle $P$ of mass $5$ kg is held at rest on a rough plane inclined at $30^\circ$ to the horizontal by a light string attached to $P$. The string lies along a line of greatest slope of the plane and is directed up the plane. The coefficient of friction between $P$ and the plane is $0.4$. Take $g=9.8$ m s⁻².

   (a)

   The tension in the string is $T$ N and $P$ is on the point of slipping down the plane. Find the value of $T$.

   [6 marks]

   (b)

   Find the greatest tension in the string for which $P$ remains in equilibrium, and hence state the range of values of $T$ for which $P$ is in equilibrium.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   Two forces act on a particle: $\mathbf{P}=(5\mathbf{i}+12\mathbf{j})$ N and $\mathbf{Q}=(-2\mathbf{i}+4\mathbf{j})$ N, where $\mathbf{i}$ and $\mathbf{j}$ are perpendicular unit vectors.

   (a)

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

   [1 mark]

   • A$17$ N
   • B$7$ N
   • C$13$ N
   • D$169$ N

   (b)

   Find the magnitude of the resultant of $\mathbf{P}$ and $\mathbf{Q}$.

   [1 mark]

   • A$16.3$ N
   • B$17.5$ N
   • C$8.5$ N
   • D$16$ N

   (c)

   A third force $\mathbf{R}$ is applied so that the particle is in equilibrium. Find $\mathbf{R}$.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   A hanging basket of weight $90$ N is suspended in equilibrium from a hook by two light strings. The strings have equal tension $T$ N and each makes an angle of $60^\circ$ with the vertical.

   (a)

   Find the value of $T$.

   [1 mark]

   • A$45$
   • B$52.0$
   • C$180$
   • D$90$

   (b)

   Find the horizontal component of the tension in one string.

   [1 mark]

   • A$45$ N
   • B$77.9$ N
   • C$90$ N
   • D$156$ N

   (c)

   Each string breaks if its tension exceeds $60$ N. The angle of each string to the vertical remains $60^\circ$. Find the greatest weight the basket can have without a string breaking.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   A small ball of mass $2$ kg is attached to one end of a light inextensible string. The other end of the string is fixed. The ball is held in equilibrium by a horizontal force of magnitude $H$ N so that the string makes an angle of $35^\circ$ with the vertical. Take $g=9.8$ m s⁻².

   (a)

   Find the value of $H$.

   [3 marks]

   (b)

   The horizontal force is doubled in magnitude and the ball comes to rest in a new equilibrium position. Find the new angle between the string and the vertical, and the new tension in the string.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   A particle $P$ of mass $3$ kg is held in equilibrium, hanging from two light inextensible strings $AP$ and $BP$. The other ends $A$ and $B$ are attached to a horizontal ceiling, on opposite sides of $P$. The string $AP$ makes an angle of $30^\circ$ with the ceiling and $BP$ makes an angle of $50^\circ$ with the ceiling. The strings and $P$ lie in the same vertical plane. Take $g=9.8$ m s⁻².

   (a)

   Find the tension in each string.

   [6 marks]

   (b)

   The same two strings are used, at the same angles, to support a different particle of mass $m$ kg. String $AP$ breaks if its tension exceeds $30$ N and string $BP$ breaks if its tension exceeds $36$ N. Find the greatest value of $m$ for which neither string breaks, stating which string would break first.

   [6 marks]

   Total for question 8: 12 marks

End of questions