M5S7: Quantitative Analysis of Circular Motion

solve problems, model and make quantitative predictions about objects executing uniform circular motion in a variety of situations, using the following relationships:
 Tangential velocity v is defined as the total circumference of circular path divided by the total time taken to complete one revolution T, which is also known as the period of circular motion.
`v=(2pir)/T`
 Angular velocity w is defined as the total angle of one revolution, 2p, divided by the period, T, of circular motion. Its SI unit is radians per second (rad s^{1})
Alternatively, angular velocity is also defined as the angle completed by an object in circular motion in time t, where t does not necessarily need to be the period of circular motion.
`omega=(Deltatheta)/t`
Figure shows the direction of tangential and angular velocity. vector points perpendicular to the v vector, can be determined by curling the right hand into a fist form with thumb pointing outwards. Curled fingers represent direction of rotation while thumb points in the direction of angular velocity.
Substituting into centripetal force equation (`F_c=(mv^2)/r`) gives `F_c=mromega^2` .
Derivation:
Practice Question 1 (NESA Sample Question)
A 15gram metal ball bearing on a string is swung around a pole in a circle of radius 0.8 m. The plane of the circular path is horizontal. The angular velocity of the motion is `4pi rad s^1`.
What is the magnitude of the centripetal force required to maintain the motion of the ball?
Practice Question 2
A particle is moving around in a circle of radius 1.5 m with a constant speed of 2 ms^{1}. Calculate its
(a) centripetal acceleration(b) angular velocity
Practice Question 3 (HSC 2013)
The diagram shows a futuristic space station designed to simulate gravity in a weightless environment.
If the space station has a diameter of 550 m, calculate the rotational speed needed to simulate 1g of gravitational acceleration.
Practice Question 4
A 15 kg weight is attached to a rope and spun in uniform circular motion with a 0.5 m radius as shown.
If the rope makes an angle of 30º with the horizontal during circular motion, calculate the angular velocity of the weight.
Solutions
Practice Question 1
`F_c=mromega^2`
Convert grams into SI unit (kg):
`F_c=(15/1000)(0.8)(4pi)^2`
`F_c=1.9` N
Practice Question 2
(a) We will use `a_c=v^2/r` to find centripetal acceleration:
(b) We will use `omega=v/r` to find angular velocity: