Free Addition of Vectors MCQs with Answers

1. Two forces of 3 N and 4 N act at right angles to each other. The magnitude of their resultant is

• A. 1 N
• B. 5 N
• C. 7 N
• D. 12 N

Explanation: For perpendicular vectors the resultant is the square root of (3 squared plus 4 squared), which is the square root of 25, giving 5 N. Adding the magnitudes to get 7 N would only be correct if the forces acted in the same direction, and 1 N is the result if they were exactly opposed. Every answer between 1 N and 7 N is possible depending on the angle.

2. Two forces of equal magnitude F act at an angle of 120 degrees to each other. Their resultant has magnitude

• A. 2F
• B. F times the square root of 3
• C. F
• D. zero

Explanation: Using the parallelogram rule, the resultant is the square root of (F squared plus F squared plus 2F squared cos 120), and since cos 120 is minus one half the last term cancels one of the others, leaving the square root of F squared, which is F. Equal vectors at 120 degrees are the arrangement in which three such vectors would sum to zero. At 0 degrees the resultant would be 2F and at 180 degrees it would be zero.

3. The minimum number of unequal coplanar forces whose vector sum can be zero is

• A. one
• B. two
• C. three
• D. four

Explanation: Two unequal forces can never cancel, since cancelling requires equal magnitudes in opposite directions. Three forces can close a triangle head to tail even when all three magnitudes differ, so three is the minimum. This is why the triangle of forces is the standard construction for a body in equilibrium under three forces.

4. The first condition of equilibrium requires that

• A. the vector sum of all forces acting on the body is zero
• B. the sum of all torques about any point is zero
• C. the body is at rest
• D. no force acts on the body at all

Explanation: With zero net force there is no acceleration of the centre of mass, which is translational equilibrium. The second condition deals with rotation and requires zero net torque, and a body needs both to be in complete equilibrium. Note that equilibrium allows motion at constant velocity, so being at rest is not required.

5. A body can be in translational equilibrium yet still start rotating if

• A. its mass is very small
• B. the net force is zero but the net torque is not
• C. the forces are all vertical
• D. it is moving at constant speed

Explanation: Two equal and opposite forces acting along different lines form a couple: they cancel as vectors, so there is no acceleration, but they produce a turning effect about the centre. This is exactly how a steering wheel is turned. Complete equilibrium therefore requires the second condition as well as the first.

6. The resultant of two vectors is maximum when the angle between them is

• A. 90 degrees
• B. 180 degrees
• C. 120 degrees
• D. 0 degrees

Explanation: Parallel vectors add directly, so the resultant equals the sum of the magnitudes, which is the largest value achievable. At 180 degrees they oppose and the resultant is the difference, the minimum. Every intermediate angle gives a resultant between those two limits, which is the useful check on any answer.

7. A ladder rests against a smooth wall. The wall exerts a force on the ladder that is

• A. along the ladder
• B. vertically upwards
• C. perpendicular to the wall
• D. parallel to the wall

Explanation: A smooth surface can exert no friction, so the only force it can apply is the normal reaction, which by definition acts at right angles to the surface. That makes the reaction from a vertical wall horizontal. The ground, being rough, supplies both a normal force and the friction that stops the ladder sliding out.

8. Three forces acting on a body are in equilibrium. When drawn head to tail they must

• A. lie along one straight line
• B. form a closed triangle
• C. all be equal in magnitude
• D. be mutually perpendicular

Explanation: A zero resultant means the head of the last vector coincides with the tail of the first, so the three vectors close a triangle. This triangle of forces lets an unknown force be found by scale drawing or by the sine rule. The three need not be equal, and they cannot be collinear unless two of them point the same way.