MIT 8.01 - Classical Mechanics: Kinematics, Dynamics & Energy Conservation Flashcards

Question 1

1D Kinematics Equations (Constant Acceleration)

Accepted Answer

For constant acceleration $a$, initial position $x_0$, and initial velocity $v_0$:
1. $v(t) = v_0 + at$
2. $x(t) = x_0 + v_0 t + \frac{1}{2}at^2$
3. $v^2 = v_0^2 + 2a(x - x_0)$

Question 2

Uniform Circular Motion Acceleration

Accepted Answer

An object moving in a circle of radius $r$ at constant speed $v$ experiences centripetal acceleration directed radially toward the center:
$$a_c = \frac{v^2}{r} = \omega^2 r$$
where $\omega$ is angular velocity.

Question 3

Newton's Second Law of Motion

Accepted Answer

The net force acting on an object equals the rate of change of its linear momentum:
$$\vec{F}_{	ext{net}} = \frac{d\vec{p}}{dt}$$
For constant mass $m$, this simplifies to standard vector form:
$$\sum \vec{F} = m\vec{a}$$

Question 4

Static vs. Kinetic Friction

Accepted Answer

Static friction opposes initial sliding and adjusts to balance applied force up to a maximum threshold:
$$f_s \le \mu_s N$$
Kinetic friction opposes ongoing relative sliding motion with fixed magnitude:
$$f_k = \mu_k N$$
Generally, $\mu_s > \mu_k$.

Question 5

Line Integral Definition of Work

Accepted Answer

Work $W$ done by a force $\vec{F}$ moving an object along a path from position $\vec{r}_1$ to $\vec{r}_2$ is defined as:
$$W = \int_{\vec{r}_1}^{\vec{r}_2} \vec{F} \cdot d\vec{r} = \int_{\vec{r}_1}^{\vec{r}_2} F \cos(	heta) \, ds$$

Question 6

Work-Energy Theorem

Accepted Answer

The total work done by all forces (conservative and non-conservative) acting on a particle equals the change in its kinetic energy $K$:
$$W_{	ext{net}} = \Delta K = K_f - K_i = \frac{1}{2}m v_f^2 - \frac{1}{2}m v_i^2$$

Question 7

Conservative Force & Potential Energy

Accepted Answer

A force is conservative if work done around any closed loop is zero. Potential energy $U$ is defined via:
$$W_{	ext{cons}} = -\Delta U$$
In continuous form, force is the negative gradient of potential energy:
$$\vec{F} = -
abla U = -\left(\frac{\partial U}{\partial x}\hat{i} + \frac{\partial U}{\partial y}\hat{j} + \frac{\partial U}{\partial z}\hat{k}ight)$$

Question 8

Conservation of Mechanical Energy

Accepted Answer

Total mechanical energy $E_{	ext{mech}} = K + U$ is conserved when only conservative forces do work ($W_{	ext{nc}} = 0$):
$$\Delta E_{	ext{mech}} = 0 \implies K_i + U_i = K_f + U_f$$

Question 9

Impulse-Momentum Theorem

Accepted Answer

Linear momentum is defined as $\vec{p} = m\vec{v}$. Impulse $\vec{J}$ is the integral of net force over time, equal to total momentum change:
$$\vec{J} = \int_{t_1}^{t_2} \vec{F}_{	ext{net}} \, dt = \Delta \vec{p} = \vec{p}_f - \vec{p}_i$$

Question 10

Conservation of Linear Momentum

Accepted Answer

For a system of particles, total linear momentum $\vec{P}_{	ext{tot}} = \sum \vec{p}_i$ is strictly conserved if net external force is zero:
$$\sum \vec{F}_{	ext{ext}} = 0 \implies \frac{d\vec{P}_{	ext{tot}}}{dt} = 0 \implies \vec{P}_{	ext{tot}} = 	ext{constant}$$

Question 11

Elastic vs. Inelastic Collisions

Accepted Answer

- **Elastic Collision**: Linear momentum and kinetic energy are both conserved ($\Delta \vec{P} = 0$, $\Delta K = 0$).
- **Inelastic Collision**: Momentum is conserved, but kinetic energy is lost to heat/deformation ($\Delta \vec{P} = 0$, $\Delta K < 0$).
- **Completely Inelastic**: Objects stick together post-collision, maximizing kinetic energy loss.

Question 12

Center of Mass (Discrete & Continuous)

Accepted Answer

- **Discrete**: $\vec{r}_{	ext{cm}} = \frac{1}{M}\sum m_i \vec{r}_i$ where total mass $M = \sum m_i$.
- **Continuous**: $\vec{r}_{	ext{cm}} = \frac{1}{M}\int \vec{r} \, dm$.
Total momentum relates via $\vec{P}_{	ext{tot}} = M \vec{v}_{	ext{cm}}$.

Question 13

Vector Definition of Torque

Accepted Answer

Torque $\vec{	au}$ about a chosen point caused by force $\vec{F}$ applied at position $\vec{r}$ is:
$$\vec{	au} = \vec{r} 	imes \vec{F}$$
Magnitude: $	au = r F \sin(	heta)$, direction determined by the right-hand rule.

Question 14

Moment of Inertia

Accepted Answer

A measure of rotational inertia about a specified axis:
- **Discrete**: $I = \sum m_i r_i^2$
- **Continuous**: $I = \int r^2 dm$
where $r$ is perpendicular distance from the axis of rotation.

Question 15

Newton's Second Law for Rotation

Accepted Answer

For fixed-axis rotation of a rigid body, net external torque equals moment of inertia times angular acceleration:
$$\sum \vec{	au}_{	ext{ext}} = I \vec{\alpha}$$
where $\vec{\alpha} = \frac{d\vec{\omega}}{dt} = \frac{d^2\vec{	heta}}{dt^2}$.

Question 16

Parallel Axis Theorem

Accepted Answer

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

Question 17

Rolling Without Slipping Condition

Accepted Answer

For a rigid round object of radius $R$ rolling without slipping:
- Velocity condition: $v_{	ext{cm}} = R\omega$
- Acceleration condition: $a_{	ext{cm}} = R\alpha$
- Total kinetic energy: $K_{	ext{tot}} = \frac{1}{2} M v_{	ext{cm}}^2 + \frac{1}{2} I_{	ext{cm}} \omega^2$

Question 18

Angular Momentum Vector Definitions

Accepted Answer

- **Single Particle**: $\vec{L} = \vec{r} 	imes \vec{p} = \vec{r} 	imes (m\vec{v})$
- **Rigid Body (Symmetric Axis)**: $\vec{L} = I \vec{\omega}$
Rate of change relates to torque: $\frac{d\vec{L}}{dt} = \vec{	au}_{	ext{net}}$.

Question 19

Conservation of Angular Momentum

Accepted Answer

Total angular momentum $\vec{L}$ is conserved when net external torque about a specified origin is zero:
$$\sum \vec{	au}_{	ext{ext}} = 0 \implies \vec{L}_i = \vec{L}_f$$

Question 20

Simple Harmonic Motion (SHM) Differential Equation

Accepted Answer

The canonical SHM differential equation is:
$$\frac{d^2 x}{dt^2} + \omega^2 x = 0$$
General solution: $x(t) = A \cos(\omega t + \phi)$, where $A$ is amplitude, $\omega$ is angular frequency, and $\phi$ is phase angle.

Question 21

Frequencies of Mass-Spring & Simple Pendulum

Accepted Answer

- **Mass-Spring**: Angular frequency $\omega = \sqrt{\frac{k}{m}}$, Period $T = 2\pi \sqrt{\frac{m}{k}}$
- **Simple Pendulum** (small angle approximation $\sin	heta \approx 	heta$): Angular frequency $\omega = \sqrt{\frac{g}{L}}$, Period $T = 2\pi \sqrt{\frac{L}{g}}$

Question 22

Newton's Law of Universal Gravitation

Accepted Answer

Gravitational force between two point masses $m_1$ and $m_2$ separated by distance $r$:
$$\vec{F}_{12} = -G \frac{m_1 m_2}{r^2} \hat{r}_{12}$$
Corresponding gravitational potential energy (setting $U(\infty) = 0$):
$$U(r) = -G \frac{m_1 m_2}{r}$$

MIT 8.01 - Classical Mechanics: Kinematics, Dynamics & Energy Conservation

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