OCR MEI M1 (Mechanics 1)

1 Fig. 2 shows two forces acting at A . The figure also shows the perpendicular unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) which are respectively horizontal and vertically upwards. The resultant of the two forces is \(\mathbf { F } \mathbf { N }\). \begin{figure}[h] \end{figure}

1. Find \(\mathbf { F }\) in terms of \(\mathbf { i }\) and \(\mathbf { j }\), giving your answer correct to three significant figures.
2. Calculate the magnitude of \(\mathbf { F }\) and the angle that \(\mathbf { F }\) makes with the upward vertical.

2 Force \(\mathbf { F }\) is \(\left( \begin{array} { l } 4
1
2 \end{array} \right) \mathrm { N }\) and force \(\mathbf { G }\) is \(\left( \begin{array} { r } - 6
2
4 \end{array} \right) \mathrm { N }\).

1. Find the resultant of \(\mathbf { F }\) and \(\mathbf { G }\) and calculate its magnitude.
2. Forces \(\mathbf { F }\), \(2 \mathbf { G }\) and \(\mathbf { H }\) act on a particle which is in equilibrium. Find \(\mathbf { H }\).

3 A box of mass 5 kg is at rest on a rough horizontal floor.

1. Find the value of the normal reaction of the floor on the box. The box remains at rest on the floor when a force of 10 N is applied to it at an angle of \(40 ^ { \circ }\) to the upward vertical, as shown in Fig. 3. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{94f23528-931c-47b6-89aa-4b6edd25cc30-2_286_470_1067_803} \captionsetup{labelformat=empty} \caption{Fig. 3}
   \end{figure}
2. Draw a diagram showing all the forces acting on the box.
3. Calculate the new value of the normal reaction of the floor on the box and also the frictional force.

4 Fig. 4 shows forces of magnitudes 20 N and 16 N inclined at \(60 ^ { \circ }\). \begin{figure}[h] \end{figure}

1. Calculate the component of the resultant of these two forces in the direction of the 20 N force.
2. Calculate the magnitude of the resultant of these two forces. These are the only forces acting on a particle of mass 2 kg .
3. Find the magnitude of the acceleration of the particle and the angle the acceleration makes with the 20 N force.

5 A particle is in equilibrium when acted on by the forces \(\left( \begin{array} { r } x
- 7
z \end{array} \right) , \left( \begin{array} { r } 4
y
- 5 \end{array} \right)\) and \(\left( \begin{array} { r } 5
4
- 7 \end{array} \right)\), where the units are newtons.

1. Find the values of \(x , y\) and \(z\).
2. Calculate the magnitude of \(\left( \begin{array} { r } 5
   4
   - 7 \end{array} \right)\).

6 A small box B of weight 400 N is held in equilibrium by two light strings AB and BC . The string \(B C\) is fixed at \(C\). The end \(A\) of string \(A B\) is fixed so that \(A B\) is at an angle \(\alpha\) to the vertical where \(\alpha < 60 ^ { \circ }\). String BC is at \(60 ^ { \circ }\) to the vertical. This information is shown in Fig. 5. \begin{figure}[h] \end{figure}

1. Draw a labelled diagram showing all the forces acting on the box.
2. In one situation string AB is fixed so that \(\alpha = 30 ^ { \circ }\). By drawing a triangle of forces, or otherwise, calculate the tension in the string BC and the tension in the string AB .
3. Show carefully, but briefly, that the box cannot be in equilibrium if \(\alpha = 60 ^ { \circ }\) and BC remains at \(60 ^ { \circ }\) to the vertical.