Forces and Newton's laws (AS)AQA A-Level Maths: Topic test

1. 1
   A rocket sled of mass $300$ kg moves along a straight horizontal track at a constant velocity of $20$ m s⁻¹. The only horizontal forces on it are a driving force of $450$ N and a resistance, which stays constant.

   (a)

   What is the magnitude of the resistance?

   [1 mark]

   • A$450$ N
   • B$150$ N
   • C$2940$ N
   • D$0$ N

   (b)

   The driving force is increased to $600$ N. What is the acceleration of the sled?

   [1 mark]

   • A$1.5$ m s⁻²
   • B$2.0$ m s⁻²
   • C$0.5$ m s⁻²
   • D$3.5$ m s⁻²

   (c)

   The driving force is increased to $600$ N while the sled is moving at $20$ m s⁻¹. Find the distance the sled travels in the next $8$ s.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A book of mass $2$ kg rests on a horizontal table. Take $g=9.8$ m s⁻².

   (a)

   Which force forms the Newton's third law pair with the Earth's gravitational pull on the book?

   [1 mark]

   • AThe normal reaction of the table on the book
   • BThe gravitational pull of the book on the Earth
   • CThe force of the book on the table
   • DThe force of the table on the Earth

   (b)

   A person presses vertically downwards on the book with a force of $10$ N and the book stays at rest. What is the magnitude of the normal reaction of the table on the book?

   [1 mark]

   • A$9.6$ N
   • B$10$ N
   • C$19.6$ N
   • D$29.6$ N

   (c)

   The table is now moved vertically upwards with acceleration $1.5$ m s⁻², with the book remaining in contact and the person's hand removed. Find the magnitude of the normal reaction of the table on the book.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Two crates $P$ and $Q$, of masses $30$ kg and $20$ kg, lie in contact on a smooth horizontal floor. A horizontal force of $100$ N is applied to $P$, pushing $P$ against $Q$ so that the crates move together.

   (a)

   Find the acceleration of the crates and the magnitude of the force that $P$ exerts on $Q$.

   [3 marks]

   (b)

   The applied force on $P$ is changed so that the force between $P$ and $Q$ becomes $60$ N. Using Newton's third law, find the new applied force on $P$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Three horizontal forces act on a particle of mass $4$ kg on a smooth horizontal surface: $\mathbf{F}_1=(3\mathbf{i}-8\mathbf{j})$ N, $\mathbf{F}_2=(7\mathbf{i}+2\mathbf{j})$ N and $\mathbf{F}_3=(p\mathbf{i}+q\mathbf{j})$ N, where $\mathbf{i}$ and $\mathbf{j}$ are perpendicular horizontal unit vectors. The particle is at rest and stays at rest.

   (a)

   (i) Find $p$ and $q$. (ii) $\mathbf{F}_3$ is then removed. Find the acceleration as a vector and the speed of the particle $3$ s later.

   [6 marks]

   (b)

   Instead, $\mathbf{F}_3$ is $(-4\mathbf{i}+9\mathbf{j})$ N. (i) Find the magnitude of the acceleration of the particle. (ii) Find the normal reaction of the surface on the particle and, using Newton's third law, state the magnitude and direction of the force the particle exerts on the surface. Take $g=9.8$ m s⁻².

   [6 marks]

   Total for question 4: 12 marks

5. 5
   A stone of mass $0.4$ kg is thrown vertically upwards from level ground with speed $14$ m s⁻¹. Ignore air resistance and take $g=9.8$ m s⁻².

   (a)

   What is the acceleration of the stone at its highest point?

   [1 mark]

   • A$0$
   • B$9.8$ m s⁻² upwards
   • C$4.9$ m s⁻² downwards
   • D$9.8$ m s⁻² downwards

   (b)

   How long does the stone take to reach its highest point?

   [1 mark]

   • A$0.71$ s
   • B$1.43$ s
   • C$2.86$ s
   • D$14$ s

   (c)

   Find the greatest height above the ground reached by the stone.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   A hot-air balloon of mass $600$ kg, including its basket and ballast, is moving vertically downwards at a constant speed of $3$ m s⁻¹. Take $g=9.8$ m s⁻². Ignore air resistance.

   (a)

   What is the magnitude of the upward force on the balloon from the surrounding air?

   [1 mark]

   • A$1800$ N
   • B$600$ N
   • C$5880$ N
   • D$588$ N

   (b)

   A $50$ kg bag of ballast is released. The upward force from the air stays at $5880$ N. What is the acceleration of the balloon immediately afterwards?

   [1 mark]

   • A$0.89$ m s⁻² upwards
   • B$0.89$ m s⁻² downwards
   • C$0.82$ m s⁻² upwards
   • D$9.8$ m s⁻² upwards

   (c)

   After the ballast is released (upward force still $5880$ N), find the time taken for the balloon to come momentarily to rest.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   A particle $X$ of mass $3$ kg lies on a smooth horizontal table. It is attached by a light inextensible string, passing over a smooth pulley at the edge of the table, to a particle $Y$ of mass $2$ kg hanging freely. A second light inextensible string joins $Y$ to a particle $Z$ of mass $1$ kg hanging below $Y$. The system is released from rest with both strings taut. Take $g=9.8$ m s⁻².

   (a)

   Find the acceleration of the system.

   [3 marks]

   (b)

   Find the tension in each string.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   A crane lifts two crates, one above the other. Crate $A$, of mass $80$ kg, hangs from the crane's vertical cable. Crate $B$, of mass $50$ kg, hangs from $A$ by a light vertical rope. The cable and rope are light and inextensible. Take $g=9.8$ m s⁻² and ignore air resistance.

   (a)

   (i) The crates are raised at constant speed. Find the tension in the rope and in the cable. (ii) The crates are then raised with acceleration $0.6$ m s⁻². Find the tension in the rope and in the cable.

   [6 marks]

   (b)

   While the crates are moving upwards at $2$ m s⁻¹ with acceleration $0.6$ m s⁻², the rope breaks. The tension in the cable does not change at that instant. Find the acceleration of $A$ immediately after the rope breaks, and the time taken for $B$ to come momentarily to rest.

   [6 marks]

   Total for question 8: 12 marks

End of questions