… is so easy – it is just suvat again. Rotational Mechanics … is so easy – it is just suvat again.

Rotational Kinetic Energy Describe the energy changes How would you measure the kinetic energy when the ruler is at the bottom? They may suggest that they could measure the GPE and assume it is all turned in to KE When they suggest that they will measure the velocity, ask where they will measure the velocity. Someone may come up with the genius idea that they could use angular velocity as that is the same in all parts of the ruler. Then display the bob on string and show that this is easier to analyse

Rotational Kinetic Energy m v Radius = r This looks a little bit like ½mv2, but with  in place of v. If I define a new property, I = mr2 then… Worth writing the equations for Ek, v = rw and I on the real whiteboard for future reference. I is called the Moment of Inertia and is the rotational version of mass.

Moment of Inertia Define Moment of Inertia: For a small mass at a distance r from the axis For more complicated shapes, we sum together all the individual masses For example, split up a solid ruler into lots of thin strips, work out the moment of inertia of each strip and add them together! Fortunately you won’t have to do this, you will always be given the moment of inertia of each shape. Units of Moment of Inertia is kg.m2

Angular Momentum Remember I have defined I = mr2 Radius = r L = rotational version of mass x rotational version of velocity. This is called the Angular Momentum. Angular Momentum is a third property along with Energy and Momentum that is conserved in an isolated system

Ice Skaters How do Ice Skaters spin faster? As they pull their arms & legs in they are doing work, increasing the energy in the system, increasing the angular speed As they reduce radius of some mass, moment of inertia is reduced If angular momentum is (almost) conserved then angular speed will increase After you have shown the video and discussed how they have reduced I to increase angular velocity, then ask what happens to the Ek? If they reduce I by a factor of 2 then w increases by 2 and Ek also increases by a factor of 2. Where has that energy come from?

Moment of Inertia Questions A boy of mass 40kg sits on a roundabout 0.8m from the centre Calculate his moment of inertia The roundabout spins 10 times in 5 seconds Calculate the angular velocity Calculate the boy’s rotational kinetic energy Where would the boy’s 80kg father need to sit on the same roundabout to have the same rotational kinetic energy?

Moment of Inertia Questions A 0.1kg mass swings on a 1.25m string until it reaches the bottom when it hits a bar 0.5m from the mass. Calculate the moment of inertia before it hits the bar and after it hits the bar. Before it hits the bar it has an  of 4 rad.s-1 Calculate the rotational kinetic energy before it hits the bar Assuming no energy is lost in the collision, calculate the angular speed after it hits the bar. Extn: Is Angular Momentum conserved? Why? 1.25m 0.5m Could do this one or could jump straight to the first of the two exam questions. If you do you will need to: Show the two cylinders which have the same MASS. As they what they can say about the moments of inertia. Also worth showing the flywheel and explaining why they are used. Also explain the rather poor diagrams on this question. Stop for a break at this point.

Rotational SUVAT Reminder: When things rotate it is often easier to consider their rotational velocity than their linear velocity In just the same way that we defined displacement, velocity and acceleration, we can do the same for rotational (or angular) displacement, rotational velocity and rotational acceleration

Units What is the unit of angular displacement? How do you define a radian? To calculate the angle displaced in radians divide the arc length by the radius Angle = arc length m radius m Any unit of angle is strictly dimensionless! So it is OK for them to appear and disappear

Translate these equations into their equivalents They don’t get what they should do here! Show one example i.e. where it says on the linear side velocity is s/t, they should write that angular velocity is /t Translate these equations into their equivalents then fill in the units

…before you do the rest... Work Done = Force x … Distance moved in the direction of the force W = F.s.Cos  Work Done = Rotational Force x Rotational Distance moved in the direction of the Rotational Force It is not possible to apply a Torque at anything other than perpendicular to the radius, so there is no need for the Cos  term in the rotational version. W = T. Torque Angular Displacement OK? – Now just to the Equations down to Power – DON’T DO THE UNITS!

Finally, do the SUVAT equations We represent the initial and final angular velocities with 1 and 2 Then have a go at some rotational SUVAT questions

Falling Objects mgh= ½I2 + ½ mv2 For a ring, I = m r2 Gravitational Potential Energy mgh= ½I2 + ½ mv2 For a ring, I = m r2 For a solid cylinder, I = ½ m r2 The potential energy will be shared between translational and rotational kinetic energy The object with the higher Moment of inertia will have a greater fraction as rotational kinetic energy Will the solid disc arrive before, at the same time or after the Hollow one? v Kinetic Energy Rotational  Finish off the lesson with this one. I thought that this slide would effectively tell them the correct answer, but nearly 2/3 of bright E block got it wrong! Homework to hand out