Chapter 2

A force of \(30 \mathrm{~N}\) is inclined at an angle \(\theta\) to the horizontal. If its vertical component is \(18 \mathrm{~N}\), find the horizontal component and the value of \(\theta\).

The resultant of displacements \(2 \mathrm{~m}\) South, \(4 \mathrm{~m}\) West, \(5 \mathrm{~m}\) North is of magnitude: (a) \(3 \mathrm{~m}\) (b) \(7 \mathrm{~m}\) (c) \(5 \mathrm{~m}\) (d) \(\sqrt{65} \mathrm{~m}\) (e) \(11 \mathrm{~m}\).

The horizontal component of a force of \(10 \mathrm{~N}\) inclined at \(30^{\circ}\) to the vertical is: (a) \(5 \mathrm{~N}\) (b) \(5 \sqrt{3} \mathrm{~N}\) (c) \(3 \mathrm{~N}\) (d) \(\frac{10}{3} \mathrm{~N}\) (e) \(\frac{10}{\sqrt{3}} \mathrm{~N}\)

Forces represented by \(3 \mathbf{i}+5 \mathbf{j}, \mathbf{i}-2 \mathbf{j}\) and \(-3 \mathbf{i}+\mathbf{j}\) together with a fourth force F act on a particle. If the resultant force is represented by \(4 \mathrm{i}+\mathrm{j}\), find \(\mathbf{F}\).

Two vectors inclined at an angle \(\theta\) have magnitudes \(3 \mathrm{~N}\) and \(5 \mathrm{~N}\) and their resultant is of magnitude \(4 \mathrm{~N}\). The angle \(\theta\) is: (a) \(90^{\circ}\) (b) \(\arccos \frac{4}{5}\) (c) \(\arccos \frac{3}{5}\) (d) \(\arccos \frac{-3}{5}\) (e) \(60^{\circ}\)

\(\mathrm{ABCDEF}\) is a regular hexagon. Forces acting along \(\overrightarrow{\mathrm{CB}}, \overrightarrow{\mathrm{CA}}, \overrightarrow{\mathrm{CF}}\) and \(\overrightarrow{\mathrm{CD}}\) are of magnitudes \(2,4,5\) and 6 newton respectively. What is the inclination of their resultant to \(\mathrm{CF}\) ?

Two forces \(F_{1}\) and \(\mathbf{F}_{2}\) have a resultant \(\mathbf{F}_{3}\). If \(\mathrm{F}_{1}=2 \mathbf{i}-3 \mathrm{j}\) and \(\mathbf{F}_{3}=5 \mathbf{i}+4 \mathrm{j}\) then \(\mathbf{F}_{2}\) is: (a) \(7 \mathbf{i}+\mathbf{j}\) (b) \(-3 \mathrm{i}-7 \mathrm{j}\) (c) \(3 \mathrm{i}+7 \mathrm{j}\) (d) \(7 \mathrm{i}+7 \mathbf{j}\)

If a represents a velocity of \(4 \mathrm{~ms}^{-1}\) North East and \(b\) represents a velocity of \(6 \mathrm{~ms}^{-1}\) West, what velocities are represented by: (i) \(-2 \mathrm{a}\) (ii) \(\mathbf{a}+\mathbf{b}\) (iii) \(3 \mathrm{~b}-\mathrm{a}\) ?

\(\mathrm{ABC}\) is an equilateral triangle and \(\mathrm{D}\) is the mid-point of \(\mathrm{BC}\). Forces of 1,2, 4 and \(3 \sqrt{3}\) newton act along \(\overrightarrow{B C}, \overrightarrow{B A}, \overrightarrow{C A}\) and \(\overrightarrow{A D}\) respectively. Resolve each of the forces in the directions \(\mathrm{BC}\) and \(\mathrm{DA}\) and verify that the sum of the components in each direction is zero.

A force \(\mathrm{F}=3 \mathrm{i}+4 \mathrm{j}\). (a) The magnitude of the force is 5 units. (a) The magnitude of the force is 5 units. (b) The component of magnitude 3 units must be horizontal.

A force of \(2 \sqrt{2} \mathrm{~N}\) acts along the diagonal \(\mathrm{AC}\) of a square \(\mathrm{ABCD}\) and another force \(P\) acts along AD. If the resultant force is inclined at \(60^{\circ}\) to \(\mathrm{AB}\) find the value of \(P\).

A quad rilateral \(A B C D\) has opposite sides \(A B\) and DC parallel. Angle \(\mathrm{ABC}=150^{\circ}\) and angle \(\mathrm{BAD}=60^{\circ}\). Forces \(2 P, P, P, 2 P\) act along \(\overrightarrow{\mathrm{AB}}, \overrightarrow{\mathrm{BC}}, \overrightarrow{\mathrm{CD}}\), \(\overrightarrow{A D}\) respectively. Prove that the resultant has magnitude \(P(8+3 \sqrt{3})^{\frac{1}{2}}\) and find the tangent of the angle it makes with \(\mathrm{AB}\).

Calculate the magnitude of the resultant of two forces \(F_{1}\) and \(\mathbf{F}_{2}\). (a) \(\mathrm{F}_{1}=3 \mathrm{i}+7 \mathrm{j}\). b) \(F_{2}=\mathbf{i}-4 \mathbf{j}\) c) Both forces act at a point \(2 \mathrm{i}+\mathrm{j}\).

A plane lamina has perpendicular axes \(O x\) and \(O y\) marked on it, and is acted upon by the following forces: \(5 P\) in the direction \(O y\), \(4 P\) in the direction \(O x\), \(6 P\) in the direction OA where \(A\) is the point \((3 a, 4 a\) ), \(8 P\) in the direction \(\mathrm{AB}\) where \(\mathrm{B}\) is the point \((-a, a)\). Express each force in the form \(p \mathbf{i}+q \mathbf{j}\) and hence calculate the magnitude and direction of the resultant of these forces.

Six forces acting on a particle have directions parallel to the sides \(\mathrm{AB}, \mathrm{BC}\), \(C D\), DE, EF, FA of a hexagon. Find the magnitude and direction of their esultant if: a) the forces have magnitudes \(F, 2 F, 3 F, 2 F, 2 F, F\) respectively, b) the sense of each force is indicated by the order of the letters, (c) the hexagon is regular.

A speedboat which can travel at 20 knots in still water starts from the corner \(X\) of an equilateral triangle \(X Y Z\) of side 10 nautical miles and describes the complete course XYZX in the least possible time. A tide of 5 knots is running in the direction \(\overrightarrow{Z X}\). Find: (a) the speed of the boat along XY, (b) to the nearest minute the time taken by the speedboat to traverse the complete course XYZX. (1 knot is one nautical mile per hour).

Express a force \(\mathbf{F}\) in the form \(a \mathbf{i}+b \mathbf{j}\). (a) The magnitude of the force is \(5 \mathrm{~N}\). (b) The force is inclined at \(60^{\circ}\) to the horizontal. (c) \(\mathrm{j}\) is in the direction of the upward vertical.

Two vectors of equal magnitude and which are in the same direction are. equal vectors.

If \(\mathrm{F}_{1}=2 \mathrm{i}+3 \mathrm{j}\) and \(\mathrm{F}_{2}=2 \mathrm{i}-3 \mathrm{j}\) then \(\mathrm{F}_{1}\) and \(\mathbf{F}_{2}\) are equal and opposite.