Rotational Dynamics — MHT-CET Physics

Formulas (3)

• Angular Velocity, Linear Speed and Centripetal Acceleration · Circular motion
  $$\omega = 2\pi n, \qquad v = \omega r, \qquad a_c = \omega^2 r = \frac{v^2}{r}$$
• Constant Angular Acceleration · Angle in the nth second, from rest
  $$\theta_n = \frac{\alpha}{2}(2n - 1)$$
• Tangential and Centripetal Acceleration Together · Net acceleration on a circle
  $$a = \sqrt{(r\alpha)^2 + (\omega^2 r)^2}$$

Watch out for (3)

• Letting the masses matter
  → Angular Velocity, Linear Speed and Centripetal Acceleration
• Angle in the nth second versus angle in n seconds
  → Constant Angular Acceleration
• Using only the centripetal part for the net force
  → Tangential and Centripetal Acceleration Together

Formulas (3)

• The Centripetal Force and What Supplies It · Centripetal force
  $$F = \frac{mv^2}{r} = m\omega^2 r$$
• Forces at an Angle: Banked Roads, Conical Pendulums and Funnels · A tilted force turning a body
  $$\tan\theta = \frac{v^2}{rg}$$
• The Vertical Circle · Vertical circle
  $$T_{\text{bottom}} - T_{\text{top}} = 6mg, \qquad v_{\text{top,min}} = \sqrt{gr}, \quad v_{\text{bottom,min}} = \sqrt{5gr}$$

Watch out for (3)

• Adding percentages
  → The Centripetal Force and What Supplies It
• Asking for the new radius, answering the increase
  → Forces at an Angle: Banked Roads, Conical Pendulums and Funnels
• Taking the bottom tension as 5mg
  → The Vertical Circle

Formulas (3)

• Moment of Inertia of Standard Bodies · Definition
  $$I = \sum m_i r_i^2, \qquad I_{\text{ring}} = MR^2,\; I_{\text{disc}} = \tfrac{1}{2}MR^2,\; I_{\text{sphere}} = \tfrac{2}{5}MR^2$$
• Radius of Gyration · Radius of gyration
  $$I = Mk^2 \;\Rightarrow\; k = \sqrt{\frac{I}{M}}$$
• Rebuilt Bodies: Recasting, Bending and Scaling · Scaling for the same material
  $$\text{loop: } I \propto R^3, \qquad \text{disc (same thickness): } I \propto R^4$$

Watch out for (3)

• 'Same material' does not mean 'same radius'
  → Moment of Inertia of Standard Bodies
• Comparing I and reporting it as k
  → Radius of Gyration
• Keeping the radius when the material fixes the mass
  → Rebuilt Bodies: Recasting, Bending and Scaling

Formulas (4)

• The Parallel-Axis Theorem · Parallel-axis theorem
  $$I = I_{\text{cm}} + Md^2$$
• The Perpendicular-Axis Theorem · Perpendicular-axis theorem (lamina)
  $$I_z = I_x + I_y$$
• Composite Bodies: Add the Parts, Subtract the Holes · Adding about one axis
  $$I = \sum_{\text{parts}} \left(I_{\text{cm},i} + m_i d_i^2\right) \;-\; I_{\text{removed}}$$
• Where the Centre of Mass Lies · Centre of mass
  $$x_{\text{cm}} = \frac{\sum m_ix_i}{\sum m_i}$$

Watch out for (4)

• Shifting from an axis that is not through the centre of mass
  → The Parallel-Axis Theorem
• Using it on a three-dimensional body
  → The Perpendicular-Axis Theorem
• Forgetting a sphere's own moment of inertia
  → Composite Bodies: Add the Parts, Subtract the Holes
• Measuring from the wrong end
  → Where the Centre of Mass Lies

Formulas (3)

• Torque: From a Force, and From a Change in Spin · Torque
  $$\vec\tau = \vec r \times \vec F, \qquad \tau = I\alpha$$
• Angular Momentum and Its Links to Energy and Force · Angular momentum and energy
  $$L = I\omega, \qquad K = \frac{L^2}{2I}$$
• Conservation of Angular Momentum · No external torque
  $$I_1\omega_1 = I_2\omega_2, \qquad K \propto \frac{1}{I}\ \text{at fixed } L$$

Watch out for (3)

• Writing F × r
  → Torque: From a Force, and From a Change in Spin
• Equal energy means equal L
  → Angular Momentum and Its Links to Energy and Force
• Conserving kinetic energy
  → Conservation of Angular Momentum

Formulas (3)

• Rotational Kinetic Energy, and Rods That Swing or Fall · Rotational kinetic energy
  $$K_{\text{rot}} = \tfrac{1}{2}I\omega^2$$
• How a Rolling Body's Energy Splits · Rolling energy
  $$K = \tfrac{1}{2}mv^2\left(1 + \frac{k^2}{R^2}\right)$$
• Rolling Down (and Up) an Incline · Rolling on an incline
  $$a = \frac{g\sin\theta}{1 + k^2/R^2}, \qquad v = \sqrt{\frac{2gh}{1 + k^2/R^2}}$$

Watch out for (3)

• Raising the centre of mass by the whole length
  → Rotational Kinetic Energy, and Rods That Swing or Fall
• Forgetting the rotational part
  → How a Rolling Body's Energy Splits
• Using sin θ twice, or not at all
  → Rolling Down (and Up) an Incline