Kinematics of uniform circular motion
a) Define the radian and express angular displacement in radians.
Radian is the angle subtended at the center of a circle by an arc length equal to the radius of a circle is one radian.
Radian (rad) is the S.I. unit for angle, θ and it can be related to degrees in the following way. In one complete revolution, an object rotates through 360°, or 2π rad.
Angular displacement is the angle, θ, through which an object moves as it performs circular motion.
s = rθ
(θ is the angular displacement; s is the arc length; r is the radius of the circle)
b) Understand and use the concept of angular speed to solve problems.
Angular velocity (ω) of the object is the rate of change of angular displacement with respect to time.
ω = θ t = 2π/T
Uniform circular motion is the motion of a particle along a circular path with constant speed. It is accelerated motion; although speed is constant, velocity changes as direction changes.
c) Recall and use v = rω to solve problems
Linear velocity, v, of an object is its instantaneous velocity at any point in its circular path.
v = arc length/time taken = rθ/t = rω
Important points to note:
(i) The direction of the linear velocity is at a tangent to the circle described at that point. Hence it is sometimes referred to as the tangential velocity.
(ii) ω is the same for every point in the rotating object, but the linear velocity v is greater for points further from the axis.
Centripetal acceleration and centripetal force
a) Describe qualitatively motion in a curved path due to a perpendicular force, and understand the centripetal acceleration in the case of uniform motion in a circle.
A body moving in a circle at a constant speed changes velocity (since its direction changes). Thus, it always experiences an acceleration, a force and a change in momentum. The direction of resultant force (and hence acceleration) is directed towards the center.
Centripetal force is the force acting on an object in circular motion. It acts along the radius of the circular path and towards the center of the circle. It’s responsible for keeping the body moving along the circular path. It is the resultant of all forces that act on a system in circular motion.
When asked to draw a diagram showing all the forces that act on a system in circular motion, it is wrong to include a force that is labelled as ‘centripetal force‘.
b) Recall and use centripetal acceleration equations a = rω^2 and a = v^2/r.
c) Recall and use centripetal force equations F = mrω^2 and F = mv^2/r.
Equate, mv^2/r = mg,
=> v = sqrt(rg)