Free Vectors and Equilibrium MCQs with Answers

11. 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.

12. Torque is defined as

• A. force divided by distance
• B. the scalar product of force and distance
• C. the product of force and the perpendicular distance from the axis to the line of action
• D. force multiplied by time

Explanation: Only the perpendicular distance, the moment arm, counts, which is why pushing a door near its hinge is so ineffective. Written as a vector product, torque is r cross F, with magnitude rF sin theta, so a force directed straight at the axis produces no turning effect at all. Its unit is the newton metre, which is not called a joule even though the dimensions match.

13. 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.

14. A unit vector has

• A. magnitude one and specifies direction only
• B. zero magnitude
• C. magnitude equal to the vector it came from
• D. no direction

Explanation: Dividing any vector by its own magnitude gives a unit vector along the same line, which carries the direction information with the size stripped away. The unit vectors i, j and k along the three axes are used to write any vector in component form. A null vector, by contrast, has zero magnitude and an undefined direction.

15. If A is 3i + 4j, its magnitude is

• A. 7
• B. 5
• C. 12
• D. 25

Explanation: The magnitude is the square root of the sum of the squares of the components, so the square root of (9 plus 16), which is the square root of 25, that is 5. Adding the components to get 7 ignores that they are perpendicular. The value 25 is the square of the magnitude, quoted before taking the root.

16. For the vectors A equals 2i + 3j and B equals 4i minus j, the scalar product A dot B is

• A. 11
• B. 5
• C. 8
• D. minus 3

Explanation: In component form the dot product is the sum of the products of like components, so (2)(4) plus (3)(minus 1) gives 8 minus 3, which is 5. Only i with i and j with j contribute, because the unit vectors along different axes are perpendicular and their dot products vanish. The value 8 is the first term alone and 11 comes from adding rather than subtracting.

17. Which pair of quantities are both scalars?

• A. Force and acceleration
• B. Velocity and momentum
• C. Work and power
• D. Torque and displacement

Explanation: Work and power have magnitude only, with work being a dot product and power its rate of change with time. Force, acceleration, velocity, momentum, torque and displacement are all vectors. The habit worth building is to ask whether reversing the direction changes the physical meaning, since for work it does not.

18. 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.

19. 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.