Rotational Dynamics

• Moment of Inertia:

  • Any mass has a resistance to a change in velocity when subjected to a force, known as inertia.
  • The moment of inertia (I) of an object is a measure of its resistance to being rotationally accelerated about an axis. The value can be calculated for a point mass using: $I=$$mr^2$.
  • For an object of more than one point mass (extended object), you can calculate its moment of inertia by finding the sum of all individual moments of inertia from each point mass: $I =$$\Sigma mr^2$.
• Factors affecting an object's moment of inertia:

  • The object's total mass.
  • How its mass is distributed about the axis of rotation, which varies as the distance from the axis of rotation (r) is varied.
  • For example, when someone does a backflip htey may move their legs closer to their chest which decreases their moment of inertia as more of their mass is as a smaller distance from the axis of rotation (making it easier for them to rotate).
• Rotational Kinetic Energy:

  • Just like objects with linear motion, rotating objects have kinetic energy.
  • This value of total kinetic energy can be found by summing the kinetic energies of all the individual particles making up the object.
  • $E_k=$$\frac{1}{2}I\omega^2$.
• Flywheels:

  • A flywheel is a heavy metal disc that spins on an axis and has a large moment of inertia.
  • This means a large force will be required in order to be rotationally accelerated, so when it is spinning it will be difficult to stop.
  • As a flywheel is spun, the input torque is converted to rotational kinetic energy which is stored by the flywheel.
  • Flywheels can be optimised to store as much energy as possible and are known as flywheel batteries.
• Factors affecting the energy stored in a flywheel:

  • Mass of the flywheel - as its mass increases, the moment of inertia will increase. As rotational kinetic is directly proportional to the moment of inertia, this will also increase meaning more energy can be stored by the flywheel.
  • Angular speed of the flywheel - As rotational kinetic energy is proportional to the square of the angular speed, if the flywheel's angular speed increase its kinetic energy increases meaning more energy can be stored.
  • Friction - Friction between flywheel and bearings causes flywheels to lose energy.
• Uses of flywheels:

  • Regenerative Braking - Energy lost underbreaking can be used to charge a flywheel, which is then used to accelerate the vehicle later on.
  • Wind turbines - Flywheels are used to store excess power when demand is low or on windy days.
  • Smoothing torque and angular velocity - Some systems do not produce power continuously but in bursts.
  • Production processes - An electric motor is used along with a flywheel to allow the transfer of short bursts of energy. Prevents motor from 'stalling' as there are fewer large changes of power moving through it.
• Rotational Motion:

  • Angular Displacement ($\theta$) - the angle turned through in any given direction in radians.
  • Angular speed ($\omega$) - the angle an object moves through per unit time (has magnitude and direction, which can be either clockwise or anticlockwise). Units are rads$^-1$.
  • Angular acceleration (a) - The change is angular velocity over time taken. Units are rads$^-2$.
• Rotational vs Linear Equations:

  • Rotational Equations are set up in the same way as linear equations.
  • Velocity:
  • $Linear\: velocity =$$\frac{displacement}{time}$.
  • $Angular\:velocity\:(\omega) =$$\frac{angular\:displacement}{time} =$$\frac{\Delta \theta}{\Delta t}$.
  • Acceleration:
  • $Linear\:acceleration =$$\frac{velocity}{time}$.
  • $Angular\: acceleration\: (\alpha) =$$\frac{angular \:velocity}{time} =$$\frac{\Delta \omega}{\Delta t}$
• Angular Acceleration on a Graph:

  • Like linear acceleration, angular acceleration can be uniform where angular acceleration is constant.
  • When angular acceleration is uniform, a graph of angular velocity agaisnt time will be a straight line graph.
  • On these graphs, the gradient is the acceleration.
  • Non-uniform acceleration can be found by finding the gradient of a tangent at a point.
  • When angular acceleration is uniform, a graph pof angular displacement against time will show that displacement is proportional to t squared.

  
• Torque:

  • Torque (T) is the product of a force and its distance from its axis of rotation.
  • Torque often causes rotation and has the units Nm.
  • $T=$$Fr$.
  • The effect of torque can be demonstrated by a wheel and an axle. A mass attached to the axle causes a torque, so angular acceleration occurs in the wheel because there is a resultant force acting on it.
• Angular Momentum:

  • Angular momentum is the product of the moment of inertia and angular velocity of an object and its units are Nms.
  • $Angular\: momentum =$$I \omega$.
• Angular momentum:

  • When no external torque acts, the angular momentum of a system remains constant.
  • Using an ice skater spinning about a vertical axis, as their arms are moved inwards their angular velocity will increase.
  • As her radius is lower, moment of inertia is decreased. However, angular momentum is conserved so her angular velocity must increase.
• Angular Impulse:

  • This is the product of torque and its duration where the applied torque is constant, and is equal to the change in angular momentum.
  • Angular impulse = Change in angular momentum.
  • $T\Delta t=$$\Delta(I\omega)$.
  • Angular impulse can be found by calculating the area beneath a torque-time graph.

  
• Calculating Frictional Torque:

  • Apply an accelerating torque to the wheel to bring it up to a certain velocity.
  • Remove the accelerating torque, and measure the time taken for the wheel to come to rest.
  • Calculate the average deceleration by using the equation: $\alpha=$$\frac{\omega_2 -\omega_1}{t}$.
  • Finally, calculate frictional torque using the equation: $T =$$I\alpha$.
• Frictional Torque:

  • The force present that may restrict the rotation of an object.
  • Can be overcome is rotational torque is high enough.
  • It is crucial that frictional torque is minimised to reduce the amount of kinetic energy being transferred to heat and sound.
• Work Done:

  • The force causing a motion multiplied by the distance travelled.
  • Work must be done on an object in order to make it rotate, therefore to calculate work done on a rotating object you must find the product of the torque and angular displacement.
  • 𝑊=Tθ

  
• Power:

  • The rate of energy transfer the rate of doing work.
  • Power can be calculated by dividing the amount of work done by the time passed.
  • This can be used to derive an equation for work done which is the product of torque and angular velocity.

  $P =$$\frac{W}{t} =$$\frac{T\theta}{t} \: as\: \frac{\theta}{t}=$$\omega \:so\: P=$$T\omega$