Particle suspended by strings - Newton's laws and connected particles

OCR MEI M1 2008 June Q7

17 marks Moderate -0.3

What information in the question indicates that the tension in the string section CB is also 60 N ?
Show that the string sections AC and CB are equally inclined to the horizontal (so that $\alpha = \beta$ in Fig. 7.1).
Calculate the angle of the string sections AC and CB to the horizontal. In a different situation the same box is supported by two separate light strings, PC and QC, that are tied to the box at C . There is also a horizontal force of 10 N acting at C . This force and the angles between these strings and the horizontal are shown in Fig. 7.2. The box is in equilibrium. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{170edb27-324e-44df-8dc1-7d8fbad680fe-4_323_503_1649_822} \captionsetup{labelformat=empty} \caption{Fig. 7.2}
\end{figure}
Calculate the tensions in the two strings.

AQA M1 2007 June Q3

10 marks Moderate -0.8

3 A sign, of mass 2 kg , is suspended from the ceiling of a supermarket by two light strings. It hangs in equilibrium with each string making an angle of $35 ^ { \circ }$ to the vertical, as shown in the diagram. Model the sign as a particle.
\includegraphics[max width=\textwidth, alt={}, center]{81f3753c-f148-44be-8b35-0a8e531016dd-2_424_385_1790_824}

By resolving forces horizontally, show that the tension is the same in each string.
Find the tension in each string.
If the tension in a string exceeds 40 N , the string will break. Find the mass of the heaviest sign that could be suspended as shown in the diagram.

AQA Paper 2 2022 June Q18

8 marks Standard +0.3

18 An object, $O$, of mass $m$ kilograms is hanging from a ceiling by two light, inelastic strings of different lengths. The shorter string, of length 0.8 metres, is fixed to the ceiling at $A$.
The longer string, of length 1.2 metres, is fixed to the ceiling at $B$.
This object hangs 0.6 metres directly below the ceiling as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{ad6590e8-6673-45ca-bef3-a14716978827-28_252_940_667_552} 18

Show that the tension in the shorter string is over $30 \%$ more than the tension in the longer string.
18
The tension in the longer string is known to be $2 g$ newtons. Find the value of $m$.
A rough wooden ramp is 10 metres long and is inclined at an angle of $25 ^ { \circ }$ above the horizontal. The bottom of the ramp is at the point $O$. A crate of mass 20 kg is at rest at the point $A$ on the ramp.
The crate is pulled up the ramp using a rope attached to the crate.
Once in motion, the rope remains taut and parallel to the line of greatest slope of the ramp.
\includegraphics[max width=\textwidth, alt={}, center]{ad6590e8-6673-45ca-bef3-a14716978827-30_252_842_804_598}

AQA Paper 2 2023 June Q11

1 marks Easy -2.0

11 A decoration is hanging freely from a fixed point on a ceiling.
The decoration has a mass of 0.2 kilograms.
The decoration is hanging by a light, inextensible wire.
The wire is 0.1 metres long.
Find the tension in the wire. Circle your answer.
0.02 N
0.02 g N
0.2 N
0.2 g N

WJEC Unit 4 Specimen Q6

8 marks Moderate -0.3

An object of mass 4 kg is moving on a horizontal plane under the action of a constant force $4 \mathbf { i } - 12 \mathbf { j } \mathrm {~N}$. At time $t = 0 \mathrm {~s}$, its position vector is $7 \mathbf { i } - 26 \mathbf { j }$ with respect to the origin $O$ and its velocity vector is $- \mathbf { i } + 4 \mathbf { j }$.
1. Determine the velocity vector of the object at time $t = 5 \mathrm {~s}$.
2. Calculate the distance of the object from the origin when $t = 2 \mathrm {~s}$.
3. The diagram below shows an object of weight 160 N at a point $C$, supported by two cables $A C$ and $B C$ inclined at angles of $23 ^ { \circ }$ and $40 ^ { \circ }$ to the horizontal respectively.
  \includegraphics[max width=\textwidth, alt={}, center]{b35e94ab-a426-4fca-9ecb-c659e0143ed7-5_444_919_973_612}
4. Find the tension in $A C$ and the tension in $B C$.
5. State two modelling assumptions you have made in your solution.
6. The rate of change of a population of a colony of bacteria is proportional to the size of the population $P$, with constant of proportionality $k$. At time $t = 0$ (hours), the size of the population is 10 .
7. Find an expression, in terms of $k$, for $P$ at time $t$.
8. Given that the population doubles after 1 hour, find the time required for the population to reach 1 million.
9. A particle of mass 12 kg lies on a rough horizontal surface. The coefficient of friction between the particle and the surface is 0.8 . The particle is at rest. It is then subjected to a horizontal tractive force of magnitude 75 N .
  Determine the magnitude of the frictional force acting on the particle, giving a reason for your answer.
10. A body is projected at time $t = 0 \mathrm {~s}$ from a point $O$ with speed $V \mathrm {~ms} ^ { - 1 }$ in a direction inclined at an angle of $\theta$ to the horizontal.
11. Write down expressions for the horizontal and vertical components $x \mathrm {~m}$ and $y \mathrm {~m}$ of its displacement from $O$ at time $t \mathrm {~s}$.
12. Show that the range $R \mathrm {~m}$ on a horizontal plane through the point of projection is given by
$$R = \frac { V ^ { 2 } } { g } \sin 2 \theta$$
Given that the maximum range is 392 m , find, correct to one decimal place,
i) the speed of projection,
ii) the time of flight,
iii) the maximum height attained.