Physics: Classical Mechanics - Kinematics, Rotational Motion & Energy Flashcards

Question 1

Conservation of Mechanical Energy

Accepted Answer

If only conservative forces perform work on a isolated system, total mechanical energy is conserved:
$E_{mech} = K + U = 	ext{constant} \implies K_i + U_i = K_f + U_f$

Question 2

1D Kinematic Equations (Constant Acceleration)

Accepted Answer

For constant acceleration $a$, the primary kinematic equations are:
1. $v = v_0 + at$
2. $x = x_0 + v_0 t + \frac{1}{2}at^2$
3. $v^2 = v_0^2 + 2a(x - x_0)$
4. $x - x_0 = \left(\frac{v_0 + v}{2}\right) t$

Question 3

Newton's Second Law (Translational & Rotational)

Accepted Answer

Translational dynamics:
$\vec{F}_{net} = \frac{d\vec{p}}{dt} = m\vec{a}$ (for constant mass $m$)

Rotational dynamics:
$\vec{	au}_{net} = \frac{d\vec{L}}{dt} = I\vec{\alpha}$ (for constant moment of inertia $I$)

Question 4

Work-Energy Theorem

Accepted Answer

The net work done by all external forces acting on a system equals the change in its kinetic energy:
$W_{net} = \Delta K = K_f - K_i = \frac{1}{2}m v_f^2 - \frac{1}{2}m v_i^2$

Question 5

Kinetic Energy (Translational & Rotational)

Accepted Answer

Translational Kinetic Energy: $K_{trans} = \frac{1}{2}m v^2$
Rotational Kinetic Energy: $K_{rot} = \frac{1}{2}I \omega^2$

Total Kinetic Energy of a rolling rigid body:
$K_{total} = \frac{1}{2}m v_{cm}^2 + \frac{1}{2}I_{cm} \omega^2$

Question 6

Conservative Forces and Potential Energy

Accepted Answer

A force $\vec{F}$ is conservative if work done is path-independent. Potential energy $U$ is defined as:
$\Delta U = -W_{internal} = -\int_{\vec{r}_i}^{\vec{r}_f} \vec{F} \cdot d\vec{r}$

In differential form: $F_x = -\frac{dU}{dx}$ (or $\vec{F} = -
abla U$ in 3D).

Question 7

Impulse-Momentum Theorem

Accepted Answer

Impulse $\vec{J}$ is the time integral of force and equals the change in linear momentum $\vec{p}$:
$\vec{J} = \int_{t_1}^{t_2} \vec{F}(t)\, dt = \Delta \vec{p} = m\vec{v}_f - m\vec{v}_i$

Question 8

Center of Mass (Continuous System)

Accepted Answer

For a body with continuous mass distribution and mass density $\rho(\vec{r})$:
$\vec{r}_{cm} = \frac{1}{M} \int \vec{r}\, dm = \frac{1}{M} \int \vec{r} \rho(\vec{r})\, dV$
where $M = \int dm$ is the total mass.

Question 9

Torque Definition

Accepted Answer

Torque measures the rotational effectiveness of a force about an origin:
$\vec{	au} = \vec{r} 	imes \vec{F}$
Magnitude: $	au = r F \sin	heta = F r_\perp$, where $r_\perp$ is the lever arm distance.

Question 10

Moment of Inertia

Accepted Answer

Measures rotational inertia about an axis:
Discrete system: $I = \sum_{i} m_i r_i^2$
Continuous system: $I = \int r^2 dm$
where $r$ is the perpendicular distance from the axis of rotation to mass element $dm$.

Question 11

Parallel Axis Theorem

Accepted Answer

Relates the moment of inertia $I$ about any axis to $I_{cm}$ about a parallel axis through the center of mass:
$I = I_{cm} + M d^2$
where $d$ is the perpendicular distance between the two parallel axes and $M$ is total mass.

Question 12

Perpendicular Axis Theorem

Accepted Answer

Applies strictly to flat, planar 2D objects (laminae) in the $xy$-plane:
$I_z = I_x + I_y$
where $I_z$ is the moment of inertia about an axis perpendicular to the plane.

Question 13

Angular Momentum (Particle vs. Rigid Body)

Accepted Answer

Single particle: $\vec{L} = \vec{r} 	imes \vec{p} = m(\vec{r} 	imes \vec{v})$
Symmetric rigid body rotating about a fixed principal axis: $\vec{L} = I\vec{\omega}$

Question 14

Conservation of Angular Momentum

Accepted Answer

If the net external torque acting on a system about a point is zero ($\vec{	au}_{ext} = 0$), total angular momentum is conserved:
$\frac{d\vec{L}}{dt} = 0 \implies \vec{L}_i = \vec{L}_f \implies I_i \omega_i = I_f \omega_f$

Question 15

Rolling Without Slipping Condition

Accepted Answer

For a object of radius $R$ rolling without slipping along a surface:
$v_{cm} = R\omega, \quad a_{cm} = R\alpha$
The point of contact with the ground is instantaneously at rest ($v_{contact} = 0$).

Question 16

Instantaneous Power

Accepted Answer

Rate of work done or energy transfer over time:
Translational: $P = \frac{dW}{dt} = \vec{F} \cdot \vec{v}$
Rotational: $P = \vec{	au} \cdot \vec{\omega}$

Question 17

Elastic vs. Inelastic Collisions

Accepted Answer

Elastic Collision: Both total linear momentum and total kinetic energy are conserved ($K_i = K_f$).
Inelastic Collision: Momentum is conserved, but kinetic energy is lost ($K_i > K_f$). In a completely inelastic collision, the objects stick together post-collision.

Question 18

Universal Gravitational Potential Energy

Accepted Answer

For two point masses $M$ and $m$ separated by distance $r$ (with $U(\infty) = 0$):
$U(r) = -\frac{G M m}{r}$
Corresponding attractive central force: $\vec{F} = -\frac{G M m}{r^2} \hat{r}$

Question 19

Simple Harmonic Motion Energy

Accepted Answer

For a mass-spring system in SHM ($x(t) = A\cos(\omega t + \phi)$):
$K = \frac{1}{2}m v^2, \quad U = \frac{1}{2}k x^2$
Total Mechanical Energy: $E = K + U = \frac{1}{2}k A^2 = 	ext{constant}$

Question 20

Rotational Kinematic Equations (Constant Angular Acceleration)

Accepted Answer

For constant angular acceleration $\alpha$:
1. $\omega = \omega_0 + \alpha t$
2. $	heta = 	heta_0 + \omega_0 t + \frac{1}{2}\alpha t^2$
3. $\omega^2 = \omega_0^2 + 2\alpha(	heta - 	heta_0)$

Physics: Classical Mechanics - Kinematics, Rotational Motion & Energy

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