Untitled Flashcards Set

### Equations of Motion with Constant Acceleration

| Equations | Included Values |

| ---------------------------------------- | ----------------------- |

| $v_x = v_{0x} + a_xt$ | $v_x, v_{0x}, a_x$ |

| $x = x_0 + v_{0x} + \frac{1}{2} a_x t^2$ | $x_0, v_{0x}, a_x$ |

| $v_x^2 = v_{0x}^2 + 2a_x(x - x_0)$ | $x_0, v_x, v_{0x}, a_x$ |

| $x - x_0 = (\frac{v_{0x}+v_x}{2})t$ | $x_0, v_{0x}, v_x$ |

$v_f^2-v_0^2=2as$

---

### Equations of Projectile Motion

![[Pasted image 20240923121228.png]]

x = (v_0 \cos(a_0))t

y = (v_0 sin(a_0))t - \frac{1}{2}gt^2

v_x = v_0\cos(a_0)

v_y = v_0sin(a_0) - gt

---

### Uniform Circular Motion

a_{c} = \frac{v^2}{R}

a_{c} = \frac{4\pi^2R}{T^2}

---

### Newton's Laws of Motion

1. $\sum{}{}{\vec{F}}=0$

2. $\sum{}{}{\vec{F}}=m\vec{a}$

3. $\vec{F}_{\textit{A on B}} = -\vec{F}_{\textit{B on A}}$

---

### Friction

$f_k\leq{\mu_kn}$

- $n =$ normal force

- $\mu_k =$ coefficient of kinetic friction

- $f_k =$ magnitude of kinetic friction force

$f_s\leq{\mu_sn}$

- $n =$ Normal force

- $\mu_s =$ coefficient of static friction

- $f_s =$ magnitude of static friction force

---

### Dynamics of Circular Motion

![[Pasted image 20240923123552.png]]

a_{c}=\frac{v^2}{R}=\frac{4\pi^2R}{T^2}

- R - radius (m)

- T - period (s)

---

### Gravitation

Gravitation between two bodies:

F_g=\frac{Gm_1m_2}{r^2}

Weight of a body of mass m at the earth's surface:

w=F_g=\frac{Gm_Em}{R_E^2}

Acceleration due to gravity at the earth's surface:

g = \frac{Gm_E}{R_E^2}

Gravitational potential energy:

U = \frac{-Gm_Em}{r}

Acceleration:

a=\frac{GM}{r^2}=\frac{v^2}{r}

---

### Work

Constant force, straight-line displacement

W=Fs\cos(\phi)

- ***W*** - work done by force (J)

- ***F*** - Magnitude of the force (N)

- ***s*** - displacement or distance (m)

- $\phi$ - angle between the force vector and the displacement vector

Constant force, straight-line displacement:

W=\vec{F} \cdot \vec{s}

- $\vec{F}$ - force vector

- $\vec{s}$ - displacement vector

Kinetic energy:

K = \frac{1}{2}mv^2

- $K$ - Kinetic energy (J)

- can NEVER be negative

- $m$ - mass of object (g)

- $v$ - Speed or velocity of object (m/s)

Work-energy theorem:

\begin{align}

&W_{tot}=K_2-K_1=\Delta{K} \\

&W_{conservative} + W_{non-conservative} = \Delta K \\

&W_{nc} = \Delta K - W_c = \Delta K + \Delta U = \Delta E

\end{align}

- $W_{tot}$ - Total work done on the object (J)

- $K_1$ - Initial kinetic energy (J)

- $K_2$ - Final kinetic energy (J)

- $\Delta K$ - Change in kinetic energy (J)

Mechanical Energy:

E = K + U

- $E$ - Mechanical Energy (J)

- $K$ - Kinetic Energy (J)

- U - Potential Energy (J)

Change in Mechanical Energy

\Delta E = W_{nc}^{int} + W_{nc}^{ext}

Work:

W=\int_{x_1}^{x_2}F_xdx

$W$ - Work (J)

$F_x$ - Component of force in the x-direction (N)

$x_1$ and $x_2$ - Initial and final x-positions (m)

Instantaneous rate at which force $\vec{F}$ does work on a particle:

P=\vec{F}\cdot\vec{v}

Gravitational potential energy:

\begin{align}

W_{grav} &= mgy_1-mgy_2 \\

&= U_{grav,1} - U_{grav,2} \\

&= -\Delta U_{grav}

\end{align}

Elastic potential energy:

\begin{align}

W_{el} &= \frac{1}{2}kx_1^2 - \frac{1}{2}kx_1^2 \\

&= U_{el,1} - U_{el,2} \\

&= -\Delta U_{el}

\end{align}

Conserved mechanical energy:

K_1+U_1=K_2+U_2

Nonconserved mechanical energy:

K_1+U_1+W_{other}=K_2+U_2

1D force:

\begin{align}

\vec{F}&=F(x)\hat{x} \\

F(x)&=-\frac{du}{dx}

\end{align}

Average power:

P_{av}=\frac{\Delta W}{\Delta t}

$P_{av}$ - Average power (W)

$\Delta W$ - Work done (J)

$\Delta t$ - Time interval over which the work is done (s)

Instantaneous power:

P=\frac{dW}{dt}

---

# Momentum, Impulse, and Collision

#### Momentum

\vec{p} = m\vec{v}

- Impulse is the product of the net force and the time period

- If the vector sum of the external forces on a system is zero, the total momentum of the system is constant.

- The total momentum of a system of particles is equal to the sum of the momentum of each individual particle

- When momentum is conserved, the center of mass is conserved

#### Impulse

\vec{J} = \sum\vec{F}\left(t_2-t_1\right)

- Impulse is the product of the net force and the time period

\vec{J} = \vec{p_2}-\vec{p_1}

- Impulse is equivalent to the change in momentum of a particle

\vec{J} = \int_{t_1}^{t_2}\sum\vec{F}\mathrm{d}x

- Equivalent to the area under the net force curve across a time period

- Net force is the rate of change of impulse

#### Collisions

![[Pasted image 20241002104709.png]]

- Elastic collisions - the total kinetic energy of the system is conserved

- Inelastic collision - the total kinetic energy of the system is less than prior to the collision

- Completely inelastic collision - colliding bodies stick together and move as one

- Momentum is always conserved unless there are novel external forces

#### Center of Mass

M\vec{v}_{cm} = m_1\vec{v}_1 + m_2\vec{v}_2 + m_3\vec{v}_3 + ... = \vec{P}

- The total momentum of the system is equal to the total mass of a system multiplied by the velocity of the center of mass of the system

\sum\vec{F}_{ext} = M\vec{a}_{cm}

- When a body or a collection of particles is acted on by external forces, the center of mass moves just as though all the mass were concentrated at that point and it were acted on by a net force equal to the sum of the external forces on the system.

- The sum of the internal forces in a closed system equals 0.

- In a closed system, the center of mass remains constant.

---

# Fluid Mechanics

\rho = \frac{m}{V}

- Density = $\frac{\text{mass}}{\text{volume}}$

- 1 atm = 101325 Pa

F = \Delta P \times A

- F - Force

- $\Delta P$ - pressure differential

- $A$ - Area

p = p_0 + \rho gh

p = p_0 + \rho g \left(y_2-y_1\right)

- $p_0=$ atmospheric pressure

- $p_0 = 1 \text{ atm} = 101325 \text{ pa}$

#### Incompressible Liquid Continuity

A_1v_1 = A_2v_2

- $A_1$ - Area 1

- $v_1$ - velocity 1

- $A_2$ - Area 2

- $v_2$ - velocity 2

#### Volume Flow Rate

\frac{dV}{dt} = Av

#### Bernoulli's Equation

p_1 + \rho gy_1 + \frac{1}{2}\rho v_1^2 = p_2 + \rho gy_2 + \frac{1}{2}\rho v_2^2

$\rho$ - fluid density

$g$ - acceleration due to gravity

$p_1$ - pressure at elevation 1

$v_1$ - velocity at elevation 1

$y_1$ - height of elevation 1

$p_2$ - pressure at elevation 2

$v_2$ - velocity at elevation 2

$y_2$ - height of elevation 2

---

# Torque

\tau=\vec{F}*r

- $\tau$ - Torque

- $\vec{F}$ - Force perpendicular to axis

- $r$ - radius

\tau=I\alpha

- $\tau$ - Torque

- $I$ - Inertia

- $\alpha$ - angular acceleration

\tau = f_sR

- $\tau$ - Torque due the the frictional force $f_s$ acting at the radius $R$

- $f_s$ - static frictional force

- $R$ - radius

W = \int_{\Theta_1}^{\Theta_2}\tau_z d\Theta

- W - Work done by Torque

- $\tau$ - Torque

- measured by angular distance

W = \tau_z \left(\Theta_1 - \Theta_2\right)=\tau_z\Delta\Theta

- W - work done by constant torque

I = mr^2

- $I$ - Inertia

- m - mass

- $r$ - distance from the axis of rotation

K_{rotational} ​= \frac{​I\omega^2}{2}

- $K_{rotational}$ - Rotational Kinetic Energy

- $I$ - Inertia

- $\omega$ - rotational velocity

I_P = I_{cm} + Md^2

- $I_P$ - Inertia about parallel axis

- $I_{cm}$ - Inertia about center of mass

- $M$ - mass

- $d$ - distance between axis

v = \omega R

- $v$ - linear velocity

- $\omega$ - rotational velocity

- $R$ - radius

a = \alpha R

- $a$ - linear acceleration

- $\alpha$ - rotational acceleration

- $R$ - radius

L=mvr

- $L$ - angular momentum

- $m$ - mass

- $r$ - radius

---

# Simple harmonic motion

In simple harmonic motion (SHM), several key formulas describe the motion, energy, and relationships between displacement, velocity, acceleration, and time. Here are the main ones:

### 1. Displacement as a Function of Time

For an object undergoing SHM:

x(t) = A \cos(\omega t + \phi)

where:

- $x(t)$ is the displacement at time $t$,

- $A$ is the amplitude (maximum displacement),

- $\omega$ is the angular frequency,

- $\phi$ is the phase constant, which depends on the initial conditions.

### 2. Angular Frequency

The angular frequency $\omega$ is related to the spring constant $k$ and mass $m$ by:

\omega = \sqrt{\frac{k}{m}}​​

For a pendulum, $\omega$ is given by:

\omega = \sqrt{\frac{g}{L}}​​

where $g$ is the acceleration due to gravity, and $L$ is the length of the pendulum.

### 3. Velocity as a Function of Time

The velocity $v(t)$ in SHM is given by:

v(t) = \frac{dx(t)}{dt​} = -A \omega \sin(\omega t + \phi)

The maximum velocity $v_{\text{max}}$​ occurs when $\cos(\omega t + \phi) = 0$:

v_{\text{max}} = A \omega

### 4. Acceleration as a Function of Time

The acceleration $a(t)$ is:

a(t) = -A \omega^2 \cos(\omega t + \phi)

The maximum acceleration $a_{\text{max}}$​ occurs at the extreme points of displacement:

a_{\text{max}} = A \omega^2

### 5. Period and Frequency

The period $T$ is the time taken for one complete cycle of motion and is given by:

T = \frac{2\pi}{\omega}

The frequency $f$, which is the number of cycles per second, is the reciprocal of the period:

f = \frac{1}{T} = \frac{\omega}{2\pi}

### 6. Total Mechanical Energy

The total mechanical energy $E$ in SHM is constant and is given by:

E = \frac{1}{2} k A^2 = \frac{1}{2} m \omega^2 A^2

This energy is conserved and is split between potential and kinetic energy during the motion.

### 7. Potential Energy

The potential energy $U$ at a displacement $x$ is:

U = \frac{1}{2} k x^2

### 8. Kinetic Energy

The kinetic energy $K$ is:

K = \frac{1}{2} m v^2 = \frac{1}{2} m \omega^2 (A^2 - x^2)

where $v$ is the velocity at position $x$.

These formulas collectively describe the oscillatory motion and the distribution of energy in a simple harmonic oscillator.

### 9. Springs in a series

When two springs with spring constants $k_1$​ and $k_2$​ are connected in series, they act like a single spring with an effective spring constant, $k_{\text{eff}}$​, given by:

\frac{1}{k_{\text{eff}}} = \frac{1}{k_1} + \frac{1}{k_2}

or equivalently,

k_{\text{eff}} = \frac{k_1 k_2}{k_1 + k_2}​​

#### Explanation

For springs in series, the total extension xxx is the sum of the individual extensions x1x_1x1​ and $x_2$ of each spring:

x = x_1 + x_2

According to Hooke's law, each spring's extension can be written as:

- $x_1 = \frac{F}{k_1}$​

- $x_2 = \frac{F}{k_2}$​

where $F$ is the force applied to the springs.

Substituting these into the total extension, we get:

x = \frac{F}{k_1} + \frac{F}{k_2} = F \left( \frac{1}{k_1} + \frac{1}{k_2} \right)

Thus, the effective spring constant $k_{\text{eff}}$​ for the system is:

k_{\text{eff}} = \frac{F}{x} = \frac{1}{\frac{1}{k_1} + \frac{1}{k_2}} = \frac{k_1 k_2}{k_1 + k_2}​​

#### Extension to Multiple Springs in Series

For more than two springs in series, the effective spring constant can be found as:

\frac{1}{k_{\text{eff}}} = \frac{1}{k_1} + \frac{1}{k_2} + \frac{1}{k_3} + \dots

### 10. Springs in Parallel

When two or more springs with spring constants $k_1$​, $k_2$, $k_3$, etc.​ are connected in parallel, they act like a single spring with an effective spring constant, $k_{\text{eff}}$​, given by:

k_{eff} = k_1 + k_2 + k_3...

### 11. Opposing Springs

When two springs with spring constants $k_1$​ and $k_2$ are connected to opposite side of a mass, they act like a single spring with an effective spring constant, $k_{\text{eff}}$​, given by:

k_{eff} = |k_1 - k_2|

# Formula Cheat Sheet

## Simple Harmonic Motion (SHM)

1. Displacement as a function of time:

x(t) = A \cos(\omega t + \phi)

- $A$: Amplitude

- $\omega$: Angular frequency $\left(\omega = 2\pi f = \sqrt{\frac{k}{m}}\right)$

- $\phi$: Phase constant

2. Velocity:

v(t) = -\omega A \sin(\omega t + \phi)

3. Acceleration:

a(t) = -\omega^2 x(t)

4. Maximum Speed:

v_{\text{max}} = \omega A

5. Maximum Acceleration:

a_{\text{max}} = \omega^2 A

6. Total Energy in SHM:

E = \frac{1}{2}kA^2 = \frac{1}{2}mv_{\text{max}}^2

7. Angular Frequency:

- For a spring:

\omega = \sqrt{\frac{k}{m}}

- For a pendulum (small angles):

\omega = \sqrt{\frac{g}{L}}

---

## Standing Waves

1. Wave on a String (Both Ends Fixed):

- Wavelength:

\lambda_n = \frac{2L}{n}, \quad n = 1, 2, 3, \dots

- Frequency:

f_n = \frac{n}{2L} \sqrt{\frac{T}{\mu}}, \quad \mu = \frac{m}{L}

2. Wave on a String (One End Fixed, One End Free):

- Wavelength:

\lambda_n = \frac{4L}{n}, \quad n = 1, 3, 5, \dots

3. Speed of a Wave on a String:

v = \sqrt{\frac{T}{\mu}}

---

## Pendulum

1. Restoring Torque:

\tau = -MgL \sin\theta

For small angles:

\tau \approx -MgL \theta

2. Moment of Inertia for a Simple Pendulum:

I = ML^2

3. Angular Frequency:

\omega = \sqrt{\frac{g}{L}}

4. Maximum Angular Speed:

\omega_{\text{max}} = \sqrt{\frac{g}{L}} \theta_0

---

## Sound and Waves

1. Sound Intensity and Sound Level:

- Intensity:

I = \frac{P}{4\pi r^2}

$P$ - Power of the source

- Sound level (in decibels):

\beta = 10 \log_{10} \left(\frac{I}{I_0}\right)

2. Pressure Amplitude:

p_{\text{max}} = \sqrt{2} p_{\text{rms}}

I=\frac{p_{max}^2​​}{2\rho v} $p_{max}$ - Maximum amplitude of the pressure wave

3. Standing Sound Waves:

- Open pipe:

\lambda_n = \frac{2L}{n}, \quad n = 1, 2, 3, \dots

- Ends are antinodes

- Closed pipe:

\lambda_n = \frac{4L}{n}, \quad n = 1, 3, 5, \dots

- Ends are nodes

4. Beat Frequency:

f_{\text{beat}} = |f_1 - f_2|

# Standing Waves from Two Waves Added Together

## Formation of Standing Waves

1. Wave Equations:

Consider two sinusoidal waves traveling in opposite directions:

- Wave 1 (traveling to the right):

y_1(x, t) = A \sin(kx - \omega t)

- Wave 2 (traveling to the left):

y_2(x, t) = A \sin(kx + \omega t)

2. Superposition:

The total displacement is the sum of the two waves:

y(x, t) = y_1(x, t) + y_2(x, t)

3. Using Trigonometric Identity:

Using the identity:

\sin(a) + \sin(b) = 2 \sin\left(\frac{a + b}{2}\right) \cos\left(\frac{a - b}{2}\right)

Substituting $kx - \omega t$ for $a$ and $kx + \omega t$ for $b$, we get:

y(x, t) = 2A \sin(kx) \cos(\omega t)

---

## Resulting Standing Wave

The superposition forms a standing wave:

y(x, t) = 2A \sin(kx) \cos(\omega t)

- Amplitude: $2A$, the maximum amplitude of the standing wave.

- $\sin(kx)$: Determines the spatial variation (nodes and antinodes).

- $\cos(\omega t)$: Describes the oscillation in time.

---

## Conditions for Standing Waves

1. Same Wavelength and Frequency:

The two waves must have the same:

- Wavelength ($\lambda = \frac{2\pi}{k}$)

- Frequency ($f = \frac{\omega}{2\pi}$)

2. Opposite Directions:

The waves must travel in opposite directions ($+x$ and $-x$).

3. Amplitude Matching:

The amplitudes of the two waves must be equal to produce perfect nodes and antinodes.

---

## Node and Antinode Positions

1. Nodes (Points of Zero Displacement):

At nodes, $\sin(kx) = 0$, so:

kx = n\pi, \quad n = 0, 1, 2, \dots

Substituting $k = \frac{2\pi}{\lambda}$:

x = \frac{n\lambda}{2}

Nodes occur at intervals of $\frac{\lambda}{2}$.

2. Antinodes (Points of Maximum Displacement):

At antinodes, $\sin(kx) = \pm 1$, so:

kx = \frac{\pi}{2} + n\pi, \quad n = 0, 1, 2, \dots

Substituting $k = \frac{2\pi}{\lambda}$:

x = \frac{\lambda}{4} + n\frac{\lambda}{2}

Antinodes occur halfway between nodes.

---

## Key Takeaways

1. Standing Waves:

- Form when two waves with the same wavelength, frequency, and amplitude travel in opposite directions and interfere constructively and destructively.

- The result is a stationary pattern with:

- Nodes: Points of zero displacement (e.g., $x = 0, \frac{\lambda}{2}, \lambda, \dots$).

- Antinodes: Points of maximum displacement (e.g., $x = \frac{\lambda}{4}, \frac{3\lambda}{4}, \dots$).

2. Amplitude:

- The standing wave's amplitude is $2A$, double the amplitude of each traveling wave.

3. Node Spacing:

- Nodes are separated by $\frac{\lambda}{2}$.

4. Antinode Spacing:

- Antinodes are separated by $\frac{\lambda}{2}$, but they alternate with nodes (halfway in between).

---

## Practical Example

### Example:

- Two waves with $A = 1 \, \text{m}$, $\lambda = 2 \, \text{m}$, and $f = 5 \, \text{Hz}$ traveling in opposite directions.

1. Resulting Standing Wave:

y(x, t) = 2 \sin\left(\frac{\pi x}{1}\right) \cos(10\pi t)

2. Node Positions:

x = 0, 1, 2, \dots \, \text{(multiples of } \frac{\lambda}{2} \text{)}

3. Antinode Positions:

x = 0.5, 1.5, 2.5, \dots \, \text{(odd multiples of } \frac{\lambda}{4} \text{)}

This pattern repeats spatially and oscillates temporally.

---

## Energy in Rolling Motion

1. Kinetic Energy:

KE = \frac{1}{2}mv^2 + \frac{1}{2}I\omega^2

2. Moment of Inertia:

- For a disk:

I = \frac{1}{2}MR^2

- For a hoop:

I = MR^2

3. Rolling Without Slipping:

v = R\omega