Mechanics (A2): forces, friction and momentsAQA A-Level Maths: Topic test

1. 1
   Two forces, of magnitudes $10$ N and $8$ N, act at a point $O$. The angle between the lines of action of the two forces is $60^\circ$.

   (a)

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

   [1 mark]

   • A$15.6$ N
   • B$12.8$ N
   • C$9.17$ N
   • D$18.0$ N

   (b)

   What is the angle between the resultant and the $10$ N force?

   [1 mark]

   • A$30.0^\circ$
   • B$34.7^\circ$
   • C$26.3^\circ$
   • D$63.7^\circ$

   (c)

   The $8$ N force is reversed in direction. Find the magnitude of the new resultant.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A block $A$ of mass $6$ kg rests on a rough horizontal table. The coefficient of friction between $A$ and the table is $0.4$. A light inextensible string attached to $A$ passes over a smooth light pulley at the edge of the table and carries a particle $B$ of mass $4$ kg hanging freely. The system is released from rest with the string taut, and $A$ moves towards the pulley. Take $g=9.8$ m s$^{-2}$.

   (a)

   What is the magnitude of the friction force acting on $A$ while it is moving?

   [1 mark]

   • A$39.2$ N
   • B$58.8$ N
   • C$15.7$ N
   • D$23.5$ N

   (b)

   What is the acceleration of the system?

   [1 mark]

   • A$3.92$ m s$^{-2}$
   • B$1.57$ m s$^{-2}$
   • C$2.35$ m s$^{-2}$
   • D$9.80$ m s$^{-2}$

   (c)

   The acceleration of the system is $1.57$ m s$^{-2}$. Find the tension in the string.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A uniform rod $AB$ of length $2.4$ m and mass $9$ kg is hinged at $A$ to a vertical wall. The rod is held in horizontal equilibrium by a light string attached to $B$ and to the wall at a point $C$ vertically above $A$. The string makes an angle of $30^\circ$ with the rod. Take $g=9.8$ m s$^{-2}$.

   (a)

   Find the tension in the string.

   [3 marks]

   (b)

   A particle of mass $m$ kg is attached to the rod at $B$. The string can withstand a maximum tension of $150$ N. Find the greatest possible value of $m$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A uniform ladder $AB$ of length $5$ m and mass $20$ kg rests with the end $A$ on rough horizontal ground and the end $B$ against a smooth vertical wall. The ladder lies in a vertical plane perpendicular to the wall and makes an angle $\theta$ with the ground, where $\sin\theta=0.8$. The coefficient of friction between the ladder and the ground is $0.4$. Take $g=9.8$ m s$^{-2}$.

   (a)

   The ladder is in equilibrium with nobody on it. Find the normal reaction at $A$, the reaction at $B$ and the friction force at $A$.

   [6 marks]

   (b)

   (i) Show that the ladder does not slip when nobody is on it. (ii) A man of mass $80$ kg climbs the ladder. Find the greatest distance he can climb along the ladder from $A$ before it slips.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   A box of mass $12$ kg is pulled along rough horizontal ground by a light rope inclined at $25^\circ$ above the horizontal. The tension in the rope is $60$ N and the box moves in a straight line at constant speed. The coefficient of friction between the box and the ground is $\mu$. Take $g=9.8$ m s$^{-2}$.

   (a)

   What is the normal reaction of the ground on the box?

   [1 mark]

   • A$117.6$ N
   • B$92.2$ N
   • C$143$ N
   • D$54.4$ N

   (b)

   What is the value of $\mu$?

   [1 mark]

   • A$0.462$
   • B$1.70$
   • C$0.650$
   • D$0.590$

   (c)

   The box is now at rest, and the rope is made horizontal with the tension still $60$ N. Show that the box does not move.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   A uniform pole $AB$ of length $3$ m and weight $150$ N rests horizontally on a pivot at $C$, where $AC=0.5$ m. A load of weight $400$ N hangs from $A$. The pole is held in equilibrium by a vertical downward force $P$ acting at $B$.

   (a)

   What is the moment of the $400$ N load about $C$?

   [1 mark]

   • A$1200$ N m
   • B$400$ N m
   • C$200$ N m
   • D$1000$ N m

   (b)

   What is the value of $P$?

   [1 mark]

   • A$20$ N
   • B$80$ N
   • C$50$ N
   • D$140$ N

   (c)

   Express the unit of moment, the newton metre, in terms of the SI base units kg, m and s.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   Three forces act on a particle $P$ of mass $4$ kg that lies on a smooth horizontal surface. The forces are $\mathbf{F}_1=(5\mathbf{i}-2\mathbf{j})$ N, $\mathbf{F}_2=(-3\mathbf{i}+7\mathbf{j})$ N and $\mathbf{F}_3=(p\mathbf{i}+q\mathbf{j})$ N, where $\mathbf{i}$ and $\mathbf{j}$ are perpendicular horizontal unit vectors. The acceleration of $P$ is $(0.75\mathbf{i}-1.5\mathbf{j})$ m s$^{-2}$.

   (a)

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

   [3 marks]

   (b)

   (i) Find the magnitude of $\mathbf{F}_3$. (ii) $P$ starts from rest. Find the speed of $P$ after $6$ seconds.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   A parcel of mass $15$ kg is pulled at constant speed up a rough plane inclined at $25^\circ$ to the horizontal, by a rope parallel to a line of greatest slope. The coefficient of friction between the parcel and the plane is $0.3$. Model the parcel as a particle and take $g=9.8$ m s$^{-2}$.

   (a)

   Find the tension in the rope.

   [6 marks]

   (b)

   The rope breaks when the parcel is moving up the plane at $2.4$ m s$^{-1}$. (i) Find the deceleration of the parcel. (ii) Find the distance the parcel moves up the plane before it first comes to rest. (iii) Determine whether the parcel then stays at rest.

   [6 marks]

   Total for question 8: 12 marks

End of questions