Physics

Rotational and Circular Motion

5,901 words, 11 sections

This chapter is the physics of bodies that turn: wheels, pulleys, planets, and anything spinning about an axis. The MDCAT examines it heavily, with 239 distinct questions across 84 official papers between 2001 and 2026, set by all six boards, at a mean item difficulty of 1.69 on the bank scale. Three blocks carry that weight: angular kinematics with 116 questions, centripetal force and circular dynamics with 78, and rotational dynamics and moment of inertia with 45. A student who has mastered the material can move fluently between angles, angular velocity and angular acceleration, convert between the linear and the angular description of the same motion, run the rotational equations of motion, compute centripetal force and the banking angle of a road, and handle torque, moment of inertia and the conservation of angular momentum, which together cover almost every mark the chapter offers.

Rotation is measured not by the distance a point travels but by the angle through which a reference line turns. Place the vertex of the angle at the centre of the circle and let its two sides cut an arc from the circumference, as in the definition $\theta = s/r$ , where $s$ is the arc length and $r$ the radius. When the arc length equals the radius, $s = r$ and the angle is one radian. Because the radian is the ratio of two lengths, its units cancel and it is a dimensionless number.

Three units describe the same turning, and the conversions between them carry a steady share of the marks. One complete revolution is $360^{\circ}$ , which is $2\pi$ radians, so

$1\ \text{rev} = 360^{\circ} = 2\pi\ \text{rad}, \qquad 1^{\circ} = \frac{\pi}{180}\ \text{rad} \approx 0.0175\ \text{rad}, \qquad 1\ \text{rad} \approx 57.3^{\circ}.$

The finer value $1\ \text{rad} = 57^{\circ}18'$ turns up as a memorised figure, and a degree is simply one three-hundred-and-sixtieth of a revolution. Every conversion can be done from the single anchor that a full turn is $2\pi$ radians: half a revolution is $\pi$ , a quarter is $\pi/2$ , and three full turns are $3 \times 2\pi$ .

Angular displacement is the change in the angle of the reference line. It is a vector only for vanishingly small rotations, where it can be drawn along the axis of rotation; for a finite turn the order of the rotations matters and it behaves as a scalar, which is why the paper is careful to say "treated as vector for small rotations". By convention an anticlockwise displacement is positive and a clockwise one is negative. One board book adds a detail the others leave out: the interval between two successive appearances of the Sun overhead is the solar day, while the time referred to the Earth's rotation about its axis is called Universal Time, a distinction the DUHS papers draw.

Worked example

Convert $2\pi/5$ radians into degrees, and find the angle swept by the minute hand of a clock in one minute. For the first, multiply by $180/\pi$ : $\dfrac{2\pi}{5} \times \dfrac{180}{\pi} = \dfrac{360}{5} = 72^{\circ}$ . For the second, the minute hand completes one full $360^{\circ}$ turn in 60 minutes, so in one minute it sweeps $360/60 = 6^{\circ}$ . Sanity check: $72^{\circ}$ is a fifth of a full turn, and $2\pi/5$ is indeed a fifth of $2\pi$ , which agrees. The step students get wrong is multiplying by $\pi/180$ when going the other way; to move from radians to degrees multiply by $180/\pi$ , and to move from degrees to radians multiply by $\pi/180$ .

9 more sections in this chapter

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Introduction

1. Angular displacement, revolution and the radianRotation is measured not by the distance a point travels but by the angle through which a reference line turns. Place the vertex of the angle at the centre of t