Multiple particles with intermediate connections - Newton's laws and connected particles

CAIE M1 2022 June Q2

5 marks Moderate -0.3

2 Two particles $P$ and $Q$, of masses 0.5 kg and 0.3 kg respectively, are connected by a light inextensible string. The string is taut and $P$ is vertically above $Q$. A force of magnitude 10 N is applied to $P$ vertically upwards. Find the acceleration of the particles and the tension in the string connecting them.

CAIE M1 2021 November Q6

12 marks Standard +0.3

6
\includegraphics[max width=\textwidth, alt={}, center]{cb2cec83-6f8d-4c13-90a1-03bbf4e4452f-10_451_1315_258_415} The diagram shows a particle of mass 5 kg on a rough horizontal table, and two light inextensible strings attached to it passing over smooth pulleys fixed at the edges of the table. Particles of masses 4 kg and 6 kg hang freely at the ends of the strings. The particle of mass 6 kg is 0.5 m above the ground. The system is in limiting equilibrium.

Show that the coefficient of friction between the 5 kg particle and the table is 0.4 .
The 6 kg particle is now replaced by a particle of mass 8 kg and the system is released from rest.
Find the acceleration of the 4 kg particle and the tensions in the strings.
In the subsequent motion the 8 kg particle hits the ground and does not rebound. Find the time that elapses after the 8 kg particle hits the ground before the other two particles come to instantaneous rest. (You may assume this occurs before either particle reaches a pulley.)
If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

CAIE M1 2003 June Q5

8 marks Standard +0.3

5
\includegraphics[max width=\textwidth, alt={}, center]{cb04a09c-af23-4e9d-b3da-da9e351fe879-3_504_387_598_881}
$S _ { 1 }$ and $S _ { 2 }$ are light inextensible strings, and $A$ and $B$ are particles each of mass 0.2 kg . Particle $A$ is suspended from a fixed point $O$ by the string $S _ { 1 }$, and particle $B$ is suspended from $A$ by the string $S _ { 2 }$. The particles hang in equilibrium as shown in the diagram.

Find the tensions in $S _ { 1 }$ and $S _ { 2 }$. The string $S _ { 1 }$ is cut and the particles fall. The air resistance acting on $A$ is 0.4 N and the air resistance acting on $B$ is 0.2 N .
Find the acceleration of the particles and the tension in $S _ { 2 }$.

CAIE M1 2012 June Q5

7 marks Moderate -0.3

5
\includegraphics[max width=\textwidth, alt={}, center]{fa0e0e0d-b0a6-44e0-8b4f-4923e235c6c6-3_529_195_255_977} A block $A$ of mass 3 kg is attached to one end of a light inextensible string $S _ { 1 }$. Another block $B$ of mass 2 kg is attached to the other end of $S _ { 1 }$, and is also attached to one end of another light inextensible string $S _ { 2 }$. The other end of $S _ { 2 }$ is attached to a fixed point $O$ and the blocks hang in equilibrium below $O$ (see diagram).

Find the tension in $S _ { 1 }$ and the tension in $S _ { 2 }$. The string $S _ { 2 }$ breaks and the particles fall. The air resistance on $A$ is 1.6 N and the air resistance on $B$ is 4 N .
Find the acceleration of the particles and the tension in $S _ { 1 }$.

CAIE M1 2014 November Q5

9 marks Standard +0.3

5
\includegraphics[max width=\textwidth, alt={}, center]{ffefbc81-402f-4048-8741-23c8bae30d5a-3_250_846_260_648} A small block $B$ of mass 0.25 kg is attached to the mid-point of a light inextensible string. Particles $P$ and $Q$, of masses 0.2 kg and 0.3 kg respectively, are attached to the ends of the string. The string passes over two smooth pulleys fixed at opposite sides of a rough table, with $B$ resting in limiting equilibrium on the table between the pulleys and particles $P$ and $Q$ and block $B$ are in the same vertical plane (see diagram).

Find the coefficient of friction between $B$ and the table.
$Q$ is now removed so that $P$ and $B$ begin to move.
Find the acceleration of $P$ and the tension in the part $P B$ of the string.

Edexcel M1 2005 June Q1

6 marks Moderate -0.3

In taking off, an aircraft moves on a straight runway $A B$ of length 1.2 km . The aircraft moves from $A$ with initial speed $2 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. It moves with constant acceleration and 20 s later it leaves the runway at $C$ with speed $74 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. Find
1. the acceleration of the aircraft,
2. the distance $B C$.
3. Two small steel balls $A$ and $B$ have mass 0.6 kg and 0.2 kg respectively. They are moving towards each other in opposite directions on a smooth horizontal table when they collide directly. Immediately before the collision, the speed of $A$ is $8 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and the speed of $B$ is $2 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. Immediately after the collision, the direction of motion of $A$ is unchanged and the speed of $B$ is twice the speed of $A$. Find
4. the speed of $A$ immediately after the collision,
5. the magnitude of the impulse exerted on $B$ in the collision.
\begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{bd649c3c-6172-4522-bddc-a6d70088ef89-04_282_707_278_699}
\end{figure} A smooth bead $B$ is threaded on a light inextensible string. The ends of the string are attached to two fixed points $A$ and $C$ on the same horizontal level. The bead is held in equilibrium by a horizontal force of magnitude 6 N acting parallel to $A C$. The bead $B$ is vertically below $C$ and $\angle B A C = \alpha$, as shown in Figure 1. Given that $\tan \alpha = \frac { 3 } { 4 }$, find
the tension in the string,
the weight of the bead.

Edexcel M1 2017 June Q5

6 marks Moderate -0.3

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{c809d34e-83db-4a16-a831-001f9f36b1c3-14_346_241_262_845} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} A vertical light rod $P Q$ has a particle of mass 0.5 kg attached to it at $P$ and a particle of mass 0.75 kg attached to it at $Q$, to form a system, as shown in Figure 2. The system is accelerated vertically upwards by a vertical force of magnitude 15 N applied to the particle at $Q$. Find the thrust in the rod.

OCR MEI M1 2010 June Q4

7 marks Moderate -0.8

4 As shown in Fig. 4, boxes P and Q are descending vertically supported by a parachute. Box P has mass 75 kg . Box Q has mass 25 kg and hangs from box P by means of a light vertical wire. Air resistance on the boxes should be neglected. At one stage the boxes are slowing in their descent with the parachute exerting an upward vertical force of 1030 N on box P . The acceleration of the boxes is $a \mathrm {~m} \mathrm {~s} ^ { - 2 }$ upwards and the tension in the wire is $T \mathrm {~N}$. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{6cca1e5e-82b0-487d-8048-b9db7745dea6-3_341_364_210_1489} \captionsetup{labelformat=empty} \caption{Fig. 4}

\end{figure}

Draw a labelled diagram showing all the forces acting on box P and another diagram showing all the forces acting on box Q .
Write down separate equations of motion for box P and for box Q .
Calculate the tension in the wire.

OCR MEI M1 2013 June Q6

6 marks Moderate -0.3

6 A particle moves along a straight line through an origin O . Initially the particle is at O .
At time $t \mathrm {~s}$, its displacement from O is $x \mathrm {~m}$ and its velocity, $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$, is given by $$v = 24 - 18 t + 3 t ^ { 2 } .$$

Find the times, $T _ { 1 } \mathrm {~s}$ and $T _ { 2 } \mathrm {~s}$ (where $T _ { 2 } > T _ { 1 }$ ), at which the particle is stationary.
Find an expression for $x$ at time $t \mathrm {~s}$. Show that when $t = T _ { 1 } , x = 20$ and find the value of $x$ when $t = T _ { 2 }$. Section B (36 marks)
$7 \quad$ Abi and Bob are standing on the ground and are trying to raise a small object of weight 250 N to the top of a building. They are using long light ropes. Fig. 7.1 shows the initial situation. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{83e69140-4abf-4713-85da-922ce7530e47-4_773_1071_429_497} \captionsetup{labelformat=empty} \caption{Fig. 7.1}
\end{figure} Abi pulls vertically downwards on the rope A with a force $F$ N. This rope passes over a small smooth pulley and is then connected to the object. Bob pulls on another rope, B, in order to keep the object away from the side of the building. In this situation, the object is stationary and in equilibrium. The tension in rope B, which is horizontal, is 25 N . The pulley is 30 m above the object. The part of rope A between the pulley and the object makes an angle $\theta$ with the vertical.

OCR MEI Paper 1 2021 November Q2

3 marks Easy -1.8

2 An unmanned spacecraft has a weight of 5200 N on Earth. It lands on the surface of the planet Mars where the acceleration due to gravity is $3.7 \mathrm {~m} \mathrm {~s} ^ { - 2 }$. Calculate the weight of the spacecraft on Mars.

OCR MEI M1 Q2

21 marks Standard +0.3

2 A box of mass 8 kg slides on a horizontal table against a constant resistance of 11.2 N .

What horizontal force is applied to the box if it is sliding with acceleration of magnitude $2 \mathrm {~ms} ^ { - 2 }$ ? Fig. 7 shows the box of mass 8 kg on a long, rough, horizontal table. A sphere of mass 6 kg is attached to the box by means of a light inextensible string that passes over a smooth pulley. The section of the string between the pulley and the box is parallel to the table. The constant frictional force of 11.2 N opposes the motion of the box. A force of 105 N parallel to the table acts on the box in the direction shown, and the acceleration of the system is in that direction. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{0fbef619-ad15-4e46-be35-e17fed9952c0-2_372_878_870_683} \captionsetup{labelformat=empty} \caption{Fig. 7}
\end{figure}
What information in the question indicates that while the string is taut the box and sphere have the same acceleration?
Draw two separate diagrams, one showing all the horizontal forces acting on the box and the other showing all the forces acting on the sphere.
Show that the magnitude of the acceleration of the system is $2.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }$ and find the tension in the string. The system is stationary when the sphere is at point P . When the sphere is 1.8 m above P the string breaks, leaving the sphere moving upwards at a speed of $3 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.
(A) Write down the value of the acceleration of the sphere after the string breaks.
(B) The sphere passes through P again at time $T$ seconds after the string breaks. Show that $T$ is the positive root of the equation $4.9 T ^ { 2 } - 3 T - 1.8 = 0$.
( $C$ ) Using part ( $B$ ), or otherwise, calculate the total time that elapses after the sphere moves from P before the sphere again passes through P .

OCR MEI M1 Q4

7 marks Moderate -0.3

4 As shown in Fig. 4, boxes P and Q are descending vertically supported by a parachute. Box P has mass 75 kg . Box Q has mass 25 kg and hangs from box P by means of a light vertical wire. Air resistance on the boxes should be neglected. At one stage the boxes are slowing in their descent with the parachute exerting an upward vertical force of 1030 N on box P . The acceleration of the boxes is $a \mathrm {~m} \mathrm {~s} ^ { - 2 }$ upwards and the tension in the wire is $T \mathrm {~N}$. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{5a1895e1-abe3-4739-876a-f19458f0f6ed-3_332_358_1504_1526} \captionsetup{labelformat=empty} \caption{Fig. 4}

\end{figure}

Draw a labelled diagram showing all the forces acting on box P and another diagram showing all the forces acting on box Q .
Write down separate equations of motion for box P and for box Q .
Calculate the tension in the wire.

OCR M1 Q2

7 marks Moderate -0.3

2
\includegraphics[max width=\textwidth, alt={}, center]{4c8f0d10-ea1e-4aee-870d-71a52dd948ed-02_643_289_1475_927} Particles $A$ and $B$, of masses 0.2 kg and 0.3 kg respectively, are attached to the ends of a light inextensible string. Particle $A$ is held at rest at a fixed point and $B$ hangs vertically below $A$. Particle $A$ is now released. As the particles fall the air resistance acting on $A$ is 0.4 N and the air resistance acting on $B$ is 0.25 N (see diagram). The downward acceleration of each of the particles is $a \mathrm {~ms} ^ { - 2 }$ and the tension in the string is $T \mathrm {~N}$.

Write down two equations in $a$ and $T$ obtained by applying Newton's second law to $A$ and to $B$.
Find the values of $a$ and $T$. \section*{June 2005}