System of Particles and Rotational Motion — JEE Mains Physics

Formulas (2)

• Locating the centre of mass · Centre of mass
  $$x_{cm} = \frac{\sum m_i x_i}{\sum m_i} \qquad x_{cm} = \frac{\int x\,dm}{\int dm} \qquad x_{cm} = \frac{Mx_1 - mx_2}{M - m}$$
• Motion of the centre of mass, explosions and recoil · Velocity of the centre of mass, and Newton's second law for a system
  $$\vec v_{cm} = \frac{\sum m_i\vec v_i}{\sum m_i} \qquad M\vec a_{cm} = \vec F_{ext}$$

Watch out for (6)

• Averaging positions without the masses
  → Locating the centre of mass
• A hole's mass goes with its area
  → Locating the centre of mass
• Semicircular ring and semicircular disc differ
  → Locating the centre of mass
• Equal momenta, not equal speeds
  → Motion of the centre of mass, explosions and recoil
• Signs in an Atwood machine
  → Motion of the centre of mass, explosions and recoil
• An accelerating centre of mass need not curve
  → Motion of the centre of mass, explosions and recoil

Formulas (3)

• Rotational kinematics with constant and varying angular acceleration · Equations of rotation with constant angular acceleration
  $$\omega = \omega_0 + \alpha t \qquad \theta = \omega_0t + \tfrac{1}{2}\alpha t^{2} \qquad \omega^{2} = \omega_0^{2} + 2\alpha\theta$$
• Torque as the cross product of position and force · Torque
  $$\vec\tau = \vec r \times \vec F = \begin{vmatrix} \hat i & \hat j & \hat k \\ x & y & z \\ F_x & F_y & F_z \end{vmatrix}$$
• Equilibrium of a rigid body · Conditions for equilibrium
  $$\sum\vec F = 0 \qquad \sum\vec\tau_{\text{about any point}} = 0$$

Watch out for (7)

• rpm left unconverted
  → Rotational kinematics with constant and varying angular acceleration
• The angle in an interval is a difference
  → Rotational kinematics with constant and varying angular acceleration
• F × r has the wrong sign
  → Torque as the cross product of position and force
• Torque about a point that is not the origin
  → Torque as the cross product of position and force
• The middle term of the determinant
  → Torque as the cross product of position and force
• Forgetting the rod's own weight
  → Equilibrium of a rigid body
• Distances from the end, not from the pivot
  → Equilibrium of a rigid body

Formulas (1)

• Moment of inertia when a body is resized or reshaped · Disc in terms of density, and mass from volume
  $$I_{\text{disc}} = \tfrac{1}{2}MR^{2} = \tfrac{1}{2}\rho\pi R^{4}t \qquad M = \rho V$$

Reference tables (1)

Watch out for (5)

• Disc about a diameter is not MR²/2
  → Moments of inertia of standard bodies
• The radius of gyration is squared in I
  → Moments of inertia of standard bodies
• Read the radius off the stem
  → Moments of inertia of standard bodies
• The new piece keeps the old mass
  → Moment of inertia when a body is resized or reshaped
• I ∝ R⁴ only for the same material and thickness
  → Moment of inertia when a body is resized or reshaped

Formulas (2)

• Parallel-axis and perpendicular-axis theorems · Axis theorems
  $$I = I_{cm} + Md^{2} \qquad I_z = I_x + I_y \qquad k^{2} = k_{cm}^{2} + d^{2}$$
• Composite bodies and bodies with a piece removed · Adding parts and removing a hole
  $$I_{\text{system}} = \sum I_{\text{part}} \qquad I_{\text{rest}} = I_{\text{full}} - \left(I_{\text{hole}} + m_{\text{hole}}d^{2}\right)$$

Watch out for (6)

• The parallel-axis theorem starts at the centre of mass
  → Parallel-axis and perpendicular-axis theorems
• The perpendicular-axis theorem is for flat bodies
  → Parallel-axis and perpendicular-axis theorems
• A diameter given in place of a radius
  → Parallel-axis and perpendicular-axis theorems
• Parts about different axes
  → Composite bodies and bodies with a piece removed
• The removed piece has its own moment of inertia
  → Composite bodies and bodies with a piece removed
• Distance from the axis, not from the origin
  → Composite bodies and bodies with a piece removed

Formulas (2)

• Torque and angular acceleration about a fixed axis · Rotational second law, heavy pulley, and power
  $$\tau = I\alpha \qquad a = \frac{(m_1 - m_2)g}{m_1 + m_2 + I/R^{2}} \qquad P = \tau\omega$$
• Rotational kinetic energy and energy conservation · Rotational kinetic energy, and energy for a block on a wheel
  $$K_{\text{rot}} = \tfrac{1}{2}I\omega^{2} \qquad mgh = \tfrac{1}{2}mv^{2} + \tfrac{1}{2}I\omega^{2},\ v = \omega R$$

Watch out for (6)

• T = mg for an accelerating block
  → Torque and angular acceleration about a fixed axis
• Equal tensions over a heavy pulley
  → Torque and angular acceleration about a fixed axis
• Angular speed in rpm
  → Torque and angular acceleration about a fixed axis
• Leaving out the block's kinetic energy
  → Rotational kinetic energy and energy conservation
• A rod's centre of mass falls half as far
  → Rotational kinetic energy and energy conservation
• Moment of inertia about the pivot
  → Rotational kinetic energy and energy conservation

Formulas (2)

• Angular momentum of a particle · Angular momentum of a particle, and of a projectile at its highest point
  $$\vec L = \vec r \times m\vec v,\ |\vec L| = mvd \qquad L_{\text{top}} = \frac{mu^{3}\sin^{2}\theta\cos\theta}{2g}$$
• Conservation of angular momentum · Conservation of angular momentum, and energy lost on locking
  $$I_1\omega_1 = I_2\omega_2 \qquad \Delta K = \frac{I_1I_2(\omega_1 - \omega_2)^{2}}{2(I_1 + I_2)}$$

Watch out for (6)

• Distance to the point, not to the line
  → Angular momentum of a particle
• Angular momentum about a moving body
  → Angular momentum of a particle
• A projectile's L is not conserved about the launch point
  → Angular momentum of a particle
• Kinetic energy is not conserved when bodies stick
  → Conservation of angular momentum
• Day length goes with the radius squared
  → Conservation of angular momentum
• About which point is L conserved?
  → Conservation of angular momentum

Formulas (1)

• The rolling condition and the speeds of points on a rolling body · Rolling without slipping
  $$v_{cm} = \omega R \qquad v_P = \omega\, r_P \quad (r_P = \text{distance of P from the contact point})$$

Reference tables (1)

Watch out for (4)

• The top of a rolling wheel moves at 2v
  → The rolling condition and the speeds of points on a rolling body
• Friction acts only while the body slips
  → The rolling condition and the speeds of points on a rolling body
• ½mv² alone for a rolling body
  → Kinetic energy of rolling bodies
• Share of the total, or ratio of the parts
  → Kinetic energy of rolling bodies

Formulas (2)

• Speed and height of a body rolling on a slope · Speed at the foot of a slope, and height climbed
  $$v = \sqrt{\frac{2gh}{1 + k^{2}/R^{2}}} \qquad h_{\max} = \frac{v^{2}}{2g}\left(1 + \frac{k^{2}}{R^{2}}\right)$$
• Acceleration and friction of a rolling body · Acceleration and friction when rolling down a slope
  $$a = \frac{g\sin\theta}{1 + k^{2}/R^{2}} \qquad f = \frac{mg\sin\theta\,(k^{2}/R^{2})}{1 + k^{2}/R^{2}}$$

Watch out for (6)

• The slope's length is not the height
  → Speed and height of a body rolling on a slope
• A smooth slope does not stop the spin
  → Speed and height of a body rolling on a slope
• Mass and radius do not decide the race
  → Speed and height of a body rolling on a slope
• A rolling body does not accelerate at g sin θ
  → Acceleration and friction of a rolling body
• Friction pushes forward when the force is at the top
  → Acceleration and friction of a rolling body
• Needed friction is not μN
  → Acceleration and friction of a rolling body