All Free Physics MCQs with Answers

71. The scalar product of two perpendicular vectors is

• A. maximum
• B. equal to their magnitudes multiplied
• C. zero
• D. negative

Explanation: Cos 90 is zero, so the dot product vanishes whenever the vectors are at right angles, and this is the standard test for perpendicularity. It is also why no work is done by a force acting at right angles to the displacement, such as the centripetal force on a body in circular motion. The dot product is maximum when the vectors are parallel and negative when the angle exceeds 90 degrees.

72. The vector product of two parallel vectors is

• A. zero
• B. maximum and equal to AB
• C. equal to AB cos theta
• D. a scalar

Explanation: The cross product has magnitude AB sin theta, and sin 0 is zero, so parallel vectors give a null vector. It is maximum for perpendicular vectors, which is the opposite of the dot product and the easiest way to keep the two straight. The result of a cross product is always a vector, never a scalar.

73. The direction of the vector product A cross B is given by

• A. the direction of A
• B. the direction of B
• C. the bisector of the angle between A and B
• D. the right hand rule, perpendicular to the plane containing A and B

Explanation: Curling the fingers of the right hand from A to B points the thumb along A cross B, which is perpendicular to both vectors. Because the sense reverses when the order is swapped, B cross A equals minus A cross B, so the cross product is not commutative. Torque and magnetic force are both defined this way.

74. 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.

75. 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.

76. 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.

77. 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.

78. 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.

79. 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.

80. 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.