Bearing and compass direction problems - Newton's laws and connected particles

Edexcel M1 2023 January Q6

8 marks Standard +0.3

6. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{84c0eead-0a87-4d87-b33d-794a94bb466c-18_502_1429_280_319} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} A boat is pulled along a river at a constant speed by two ropes.
The banks of the river are parallel and the boat travels horizontally in a straight line, parallel to the riverbanks.

The tension in the first rope is 500 N acting at an angle of $40 ^ { \circ }$ to the direction of motion, as shown in Figure 3.
The tension in the second rope is $P$ newtons, acting at an angle of $\alpha ^ { \circ }$ to the direction of motion, also shown in Figure 3.
The resistance to motion of the boat as it moves through the water is a constant force of magnitude 900 N

The boat is modelled as a particle. The ropes are modelled as being light and lying in a horizontal plane. Use the model to find

the value of $\alpha$
the value of $P$

Edexcel M1 2017 October Q4

9 marks Moderate -0.3

4. Two forces $\mathbf { F } _ { 1 }$ and $\mathbf { F } _ { 2 }$ act on a particle. The force $\mathbf { F } _ { 1 }$ has magnitude 8 N and acts due east. The resultant of $\mathbf { F } _ { 1 }$ and $\mathbf { F } _ { 2 }$ is a force of magnitude 14 N acting in a direction whose bearing is $120 ^ { \circ }$. Find

the magnitude of $\mathbf { F } _ { 2 }$,
the direction of $\mathbf { F } _ { 2 }$, giving your answer as a bearing to the nearest degree.

Edexcel M1 2006 January Q4

9 marks Moderate -0.3

Two forces $\mathbf { P }$ and $\mathbf { Q }$ act on a particle. The force $\mathbf { P }$ has magnitude 7 N and acts due north. The resultant of $\mathbf { P }$ and $\mathbf { Q }$ is a force of magnitude 10 N acting in a direction with bearing $120 ^ { \circ }$. Find
1. the magnitude of $\mathbf { Q }$,
2. the direction of $\mathbf { Q }$, giving your answer as a bearing.
\begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{8d3635b1-2d01-48c1-a19b-37e44d593112-08_216_556_303_699}
\end{figure} A parcel of weight 10 N lies on a rough plane inclined at an angle of $30 ^ { \circ }$ to the horizontal. A horizontal force of magnitude $P$ newtons acts on the parcel, as shown in Figure 2. The parcel is in equilibrium and on the point of slipping up the plane. The normal reaction of the plane on the parcel is 18 N . The coefficient of friction between the parcel and the plane is $\mu$. Find
(a) the value of $P$,
(b) the value of $\mu$. The horizontal force is removed.
(c) Determine whether or not the parcel moves.

OCR M1 2010 June Q3

9 marks Moderate -0.3

3
\includegraphics[max width=\textwidth, alt={}, center]{4b703cf9-b3d3-4210-b57b-89136595f8a5-02_570_495_1114_826} Three horizontal forces of magnitudes $12 \mathrm {~N} , 5 \mathrm {~N}$, and 9 N act along bearings $000 ^ { \circ } , 150 ^ { \circ }$ and $270 ^ { \circ }$ respectively (see diagram).

Show that the component of the resultant of the three forces along bearing $270 ^ { \circ }$ has magnitude 6.5 N .
Find the component of the resultant of the three forces along bearing $000 ^ { \circ }$.
Hence find the magnitude and bearing of the resultant of the three forces.

OCR M1 2013 January Q1

5 marks Moderate -0.8

1 Three horizontal forces, acting at a single point, have magnitudes $12 \mathrm {~N} , 14 \mathrm {~N}$ and 5 N and act along bearings $000 ^ { \circ } , 090 ^ { \circ }$ and $270 ^ { \circ }$ respectively. Find the magnitude and bearing of their resultant.

OCR M1 2014 June Q2

7 marks Moderate -0.3

2
\includegraphics[max width=\textwidth, alt={}, center]{66eb8290-3a80-40bf-be40-a936ed7d5a1b-2_309_520_941_744} A particle rests on a smooth horizontal surface. Three horizontal forces of magnitudes $2.5 \mathrm {~N} , F \mathrm {~N}$ and 2.4 N act on the particle on bearings $\theta ^ { \circ } , 180 ^ { \circ }$ and $270 ^ { \circ }$ respectively (see diagram). The particle is in equilibrium.

Find $\theta$ and $F$. The 2.4 N force suddenly ceases to act on the particle, which has mass 0.2 kg .
Find the magnitude and direction of the acceleration of the particle.

Edexcel M1 Q1

5 marks Moderate -0.8

Two forces, both of magnitude 5 N , act on a particle in the directions with bearings $000 ^ { \circ }$ and $070 ^ { \circ }$, as shown. Calculate
1. the magnitude of the resultant force on the particle,
2. the bearing on which this resultant force acts.
3. A uniform plank $X Y$ has length 7 m and mass 2 kg . It is placed with the portion $Z Y$ in contact with a
  \includegraphics[max width=\textwidth, alt={}, center]{38e355b0-9d75-40ad-b450-bd74c5135c7f-1_149_616_843_1334}
  horizontal surface, where $Z Y = 2.8 \mathrm {~m}$. To prevent the
  \includegraphics[max width=\textwidth, alt={}, center]{38e355b0-9d75-40ad-b450-bd74c5135c7f-1_207_253_404_1505}
\section*{MECHANICS 1 (A) TEST PAPER 5 Page 2}

Edexcel M1 Q3

8 marks Moderate -0.3

3. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{60b9db45-b48e-40a1-bd22-909e11877bc3-2_442_805_1023_719} \captionsetup{labelformat=empty} \caption{Fig. 1}

\end{figure} Figure 1 shows the forces acting on a particle, $P$. These consist of a 20 N force to the South, a 6 N force to the East, an 18 N force $30 ^ { \circ }$ West of North and two unknown forces $X$ and $Y$ which act to the North-East and North respectively. Given that $P$ is in equilibrium,

show that $X$ has magnitude $3 \sqrt { } 2 \mathrm {~N}$,
find the exact value of $Y$.

WJEC Unit 2 2024 June Q6

4 marks Easy -1.2

A ship $S$ is moving with constant velocity $( 4 \mathbf { i } - 7 \mathbf { j } ) \mathrm { ms } ^ { - 1 }$, where $\mathbf { i }$ and $\mathbf { j }$ are unit vectors due east and due north respectively.
Find the speed and direction of $S$, giving the direction as a three-figure bearing, correct to the nearest degree.
The diagram below shows a forklift truck being used to raise two boxes, $P$ and $Q$, vertically. Box $Q$ rests on horizontal forks and box $P$ rests on top of box $Q$. Box $P$ has mass 25 kg and box $Q$ has mass 55 kg .
\includegraphics[max width=\textwidth, alt={}, center]{d9ef2033-bf8b-4aec-bc88-34dbc8b9c208-17_504_814_504_589}
1. When the boxes are moving upwards with uniform acceleration, the reaction of the horizontal forks on box $Q$ is 820 N . Calculate the magnitude of the acceleration.
2. Calculate the reaction of box $Q$ on box $P$ when they are moving vertically upwards with constant speed.
3. A particle, of mass 4 kg , moves in a straight line under the action of a single force $F \mathrm {~N}$, whose magnitude at time $t$ seconds is given by
$$F = 12 \sqrt { t } - 32 \text { for } t \geqslant 0 .$$
Find the acceleration of the particle when $t = 9$.
Given that the particle has velocity $- 1 \mathrm {~ms} ^ { - 1 }$ when $t = 4$, find an expression for the velocity of the particle at $t \mathrm {~s}$.
Determine whether the speed of the particle is increasing or decreasing when $t = 9$. [2]