MIT 8.01 - Classical Mechanics: Dynamics, Work, & Angular Momentum Flashcards

Question 1

Newton's Second Law (Differential Form)

Accepted Answer

In its most general form, net external force is the time rate of change of linear momentum:
$$\vec{F}_{	ext{net}} = \frac{d\vec{p}}{dt}$$
When mass $m$ is constant, this simplifies to $\vec{F}_{	ext{net}} = m\vec{a}$, where $\vec{p} = m\vec{v}$.

Question 2

Work-Kinetic Energy Theorem

Accepted Answer

The net work done by all forces acting on a particle equals the change in its kinetic energy:
$$W_{	ext{net}} = \int_{\vec{r}_1}^{\vec{r}_2} \vec{F}_{	ext{net}} \cdot d\vec{r} = \Delta K = K_f - K_i$$
where translational kinetic energy is $E_k = \frac{1}{2}mv^2$.

Question 3

Conservative Force & Potential Energy

Accepted Answer

A force is conservative if the work done around any closed loop is zero. The work done by a conservative force relates to potential energy by $W = -\Delta U$.
In 3D differential form:
$$\vec{F} = -
abla U = -\left( \frac{\partial U}{\partial x}\hat{i} + \frac{\partial U}{\partial y}\hat{j} + \frac{\partial U}{\partial z}\hat{k} ight)$$

Question 4

Impulse-Momentum Theorem

Accepted Answer

Impulse $\vec{J}$ is the integral of net force over a time interval, equal to the total change in linear momentum:
$$\vec{J} = \int_{t_1}^{t_2} \vec{F}_{	ext{net}} \, dt = \Delta \vec{p} = \vec{p}_f - \vec{p}_i$$

Question 5

Elastic vs. Perfectly Inelastic Collisions

Accepted Answer

- **Elastic Collision**: Both linear momentum $\vec{p}$ and total kinetic energy $K$ are conserved ($\Delta K = 0$).
- **Perfectly Inelastic Collision**: Momentum $\vec{p}$ is conserved, but objects stick together post-collision to share a single final velocity $v_f = \frac{m_1 v_{1i} + m_2 v_{2i}}{m_1 + m_2}$, maximizing kinetic energy loss.

Question 6

Center of Mass & System Dynamics

Accepted Answer

The position of the center of mass for a system of particles is:
$$\vec{r}_{	ext{cm}} = \frac{1}{M} \sum_{i} m_i \vec{r}_i \quad 	ext{where } M = \sum m_i$$
The overall linear momentum of the system is $\vec{P}_{	ext{tot}} = M\vec{v}_{	ext{cm}}$, and $\vec{F}_{	ext{ext, net}} = \frac{d\vec{P}_{	ext{tot}}}{dt}$.

Question 7

Definition of Torque

Accepted Answer

Torque measure of the tendency of a force to cause rotation about a point $O$:
$$\vec{	au} = \vec{r} 	imes \vec{F}$$
Magnitude: $|\vec{	au}| = r F \sin	heta = r_\perp F = r F_\perp$, where $r_\perp$ is the moment arm.

Question 8

Rotational Inertia (Moment of Inertia)

Accepted Answer

Rotational inertia measures a body's resistance to angular acceleration about an axis:
- Continuous body: $I = \int r^2 dm$
- Discrete particles: $I = \sum_{i} m_i r_i^2$
where $r$ is the perpendicular distance from the axis of rotation to the mass element.

Question 9

Parallel Axis Theorem

Accepted Answer

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

Question 10

Rotational Form of Newton's Second Law

Accepted Answer

For a rigid body rotating about a fixed axis or about its center of mass:
$$\vec{	au}_{	ext{ext, net}} = \frac{d\vec{L}}{dt} = I\vec{\alpha}$$
where $\vec{\alpha} = \frac{d\vec{\omega}}{dt}$ is angular acceleration and $I$ remains constant.

Question 11

Rotational Kinetic Energy & Work

Accepted Answer

- **Rotational Kinetic Energy**: $K_{	ext{rot}} = \frac{1}{2}I\omega^2$
- **Rotational Work**: $W = \int_{	heta_1}^{	heta_2} 	au \, d	heta$
- **Rotational Power**: $P = \vec{	au} \cdot \vec{\omega}$

Question 12

Angular Momentum Formulations

Accepted Answer

- **Point particle** about origin $O$: $\vec{L} = \vec{r} 	imes \vec{p} = m(\vec{r} 	imes \vec{v})$
- **Rigid body** rotating about a fixed symmetric axis: $L = I\omega$

Question 13

Conservation of Angular Momentum

Accepted Answer

If the net external torque acting on a system about a point $O$ is zero ($\vec{	au}_{	ext{ext, net}} = 0$):
$$\frac{d\vec{L}_{	ext{tot}}}{dt} = 0 \implies \vec{L}_{i} = \vec{L}_{f}$$
For rigid bodies with changing moment of inertia: $I_i \omega_i = I_f \omega_f$.

Question 14

Rolling Without Slipping Conditions

Accepted Answer

For a round object of radius $R$ rolling without slipping along a surface:
- Center of mass velocity: $v_{	ext{cm}} = R\omega$
- Acceleration: $a_{	ext{cm}} = R\alpha$
- Total kinetic energy: $K_{	ext{total}} = \frac{1}{2} M v_{	ext{cm}}^2 + \frac{1}{2} I_{	ext{cm}} \omega^2$

Question 15

Simple Harmonic Motion (SHM) Differential Equation

Accepted Answer

Any 1D simple harmonic oscillator obeys:
$$\frac{d^2 x}{dt^2} + \omega_0^2 x = 0$$
General solution: $x(t) = A \cos(\omega_0 t + \phi)$.
For a spring-mass system, $\omega_0 = \sqrt{\frac{k}{m}}$ and period $T = 2\pi\sqrt{\frac{m}{k}}$.

Question 16

Physical Pendulum Period

Accepted Answer

A rigid body pivoted at a point distance $d$ from center of mass undergoes small oscillation SHM:
$$\frac{d^2 	heta}{dt^2} + \left(\frac{M g d}{I}ight) 	heta = 0$$
Angular frequency: $\omega = \sqrt{\frac{M g d}{I}}$, with period $T = 2\pi \sqrt{\frac{I}{M g d}}$.

Question 17

Gyroscopic Precession

Accepted Answer

When a spinning top/gyroscope with spin angular momentum $L = I\omega$ experiences a gravitational torque $	au = M g d$, it precesses with frequency:
$$\Omega_p = \frac{	au}{L} = \frac{M g d}{I\omega}$$

Question 18

1D Elastic Collision Speed Formulae

Accepted Answer

For a 1D elastic collision between $m_1$ (velocity $v_{1i}$) and target $m_2$ at rest ($v_{2i} = 0$):
$$v_{1f} = \left( \frac{m_1 - m_2}{m_1 + m_2} \right) v_{1i}$$
$$v_{2f} = \left( \frac{2m_1}{m_1 + m_2} \right) v_{1i}$$
Relative velocity rule: $(v_{1i} - v_{2i}) = -(v_{1f} - v_{2f})$.

Question 19

Central Force Motion & Effective Potential

Accepted Answer

In central force problems (where $\vec{F} = F(r)\hat{r}$), angular momentum $L$ is conserved. Radial energy reduces to 1D motion governed by effective potential:
$$U_{	ext{eff}}(r) = U(r) + \frac{L^2}{2 m r^2}$$
where $\frac{L^2}{2 m r^2}$ is the centrifugal potential barrier.

Question 20

Work Done by Variable Force Along a Path

Accepted Answer

The work done by a variable force along path $C$ from $\vec{r}_i$ to $\vec{r}_f$:
$$W = \int_{C} \vec{F} \cdot d\vec{r} = \int_{x_i}^{x_f} F_x dx + \int_{y_i}^{y_f} F_y dy + \int_{z_i}^{z_f} F_z dz$$
Graphically, work corresponds to the area under the force vs. displacement curve.

MIT 8.01 - Classical Mechanics: Dynamics, Work, & Angular Momentum

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