- State that angular velocity, w, is rate of change of angular displacement.
- State that angular acceleration a = dw/dt and d
^{2}q/dt^{2} - State the following relationhips
w = w
_{o}+ at^{2}.... q = w_{o}t + 1/2at^{2}.... w^{2}+ w_{0}+ 2aq - Carry out calculations using the relationships in 3 above.
- State and derive the equation v = rw for a particle in circular motion.
- State and derive the equation a = ra for a particle in circular motion.
- State and derive the expressions a = v
^{2}/r or rw^{2}for the radial acceleration. - State that a central force is required to maintain circular motion
- State that the central force required depends on speed, mass and radius of rotation
- Carry out calculations us F = mv
^{2}/r.

Aids to Understanding

Introducing the Radian

Since linear velocities and accelerations, our usual terms of reference in kinematics, are defined in
terms of displacement we need to define displacement in a circular context. We will call this angular displacement, q.
It is clear that if a body travels round 1/8 circle, it subtends an angle of 45^{o}, where 1^{o} is 1/360 th part of a circle.
Although this is perfectly fine for many purposes we approach angular measurement differently. We note that the length of an arc, s, is
proportional to the angle subtended therefore it makes sense to define angular displacements, q, as the ratio of arc
length to radius; q = s/r. Thus 45^{o} is equivalent to a dimensionless number that has a value of
p/4. And we say 45^{o} ≡ p/4 radians (rad).
Since 2pr is the full circular arc (or circumference) and a body covering this length subtends an angle of
360^{0} it should be clear that 2p rad ≡ 360^{o} and p/4 rad ≡ 45^{o}.

Click for homework on Rotational Motion
Try to have this completed for the Tuesday 27th September after Rome.

In the previous topic we discussed the equations of motion in derivitave form. The motion was linear. In this topic we discuss
rotational motion and express the equations of motion in terms of angular quantities. By rotational motion we mean bodies travelling
specifically in a circle.

Equations of linear motion | Equations of rotational motion |
---|---|

s= ut + 1/2at^{2} |
q = w_{o}t + 1/2at^{2} |

v= u + at | w = w_{o} + at |

v^{2} = u^{2} +2as |
w^{2} = w_{o}^{2} +
2aq |

- Calculate w
_{o}the initial angular velocity - Calculate w
_{1}after 10 s - Calculate a,the angular acceleration
- Calculate q the angular displacement of the blades during the 10 s.

a = r(w - w

a = ra

This leads us to the following three expressions linking angular quantities to linear quantities.

- s = rq
- v = rw
- a = ra

- Calculate u, the initial velocity of the blade's tip
- Calculate v, the velocity of the blade's tip after 10 s.
- Calculate a, the acceleration of the blade's tip.
- Calculate s, the total displacement of the blade's tip during the 10 s acceleration.

Study this carefully and you should be able to see that the tension in the string is greatest at its lowest point of the flight.

If R = 0 then the car is not in contact with the bridge. The minimum speed of the car that meets this condition is the maximum speed the car can safely travel along the bridge without taking off. Rearranging mv

The context is normally a car going around a banked track or a motorcyclist tilting over as it travels around a bend. In these contexts the radial force is the component of the bodies reaction force towards the center of the track.

For this example it is the component of the reaction force, R, that constitutes the radial force. Note the vertical forces on the car are balanced but the forces towards the center (ie the radial force) is unbalanced. Look carefully at how these relationships are written above. We have ignored friction in this example.

A civil engineer wishes to bank a curve on a road so cars can travel around it with a maximum speed of 25 ms

The banking angle needs to be 28