AP Physics C: Mechanics Formulas

What is a scalar?

directionless quantity with defined magnitude

What is a vector?

quantity with defined magnitude and direction

Is distance a scalar or vector?

scalar

Is displacement a scalar or vector?

vector

What is a component of a vector?

projection of a vector onto a certain axis

Formula for magnitude of vector A?

$$A = \sqrt{A_x²+A_y²}$$

Formula for angle of a vector?

$$\theta = arctan{\frac{A_y}{A_x}}$$

Formula for average velocity?

$$v_{avg} = \frac{x_f-x_i}{t_f-t_i}$$

Formula for average acceleration?

$$a_{avg} = \frac{v_f-v_i}{t_f-t_i}$$

Is speed scalar or vector?

scalar

Is velocity scalar or vector?

vector

Formula for velocity of A relative to B?

$$v_{AB} = v_A - v_B$$

Position of a particle relative to two reference frames?

$$v_{PA} = v_{PB}+v_{BA}$$

Max height of projectile for symmetric motion

$$h = \frac{v_i²sin²(\theta)}{2g}$$

Max range for a projectile in symmetrical motion?

$$R = \frac{v_i²sin(2\theta)}{g}$$

Max time for a projectile in symmetrical motion?

$$t = \frac{2v_{y,i}}{a_y}$$

X-displacement of a projectile in parabolic motion?

$$d_x = v_icos(\theta)\Delta t$$

Centripetal acceleration formula?

$$a_c = \frac{v²}{r}$$

$F_{net}$ equals?

$$ma_{cm}$$

True/false: A particle has no net force but can have velocity

True, no net force only indicates no acceleration

What does Newton’s Third Law state?

Force exerted on object A by B is the same on object B by A

i.e., $\overrightarrow{F_{AB}} = -\overrightarrow{F_{BA}}$

Formula for STATIC friction?

$f_s \leq \mu_sF_N$
Note: There is no DEFINITE formula for static friction, it can only be less than or equal to mu times normal force

Formula for KINETIC friction?

$$f_k = \mu_kF_N$$

Does the area of contact between two objects change coefficients of friction?

no

What does static friction do?

impedes the object from moving

What is the formula for $\mu_k$?

$$\mu_k = tan(\theta)$$

For theta, use the angle where the block falls at a constant speed

What is the formula for $\mu_s$?

$$\mu_s = tan(\theta)$$

Use the angle where the block is on the verge of slipping

Spring force equation?

$$F_s = -k\Delta \overrightarrow{x}$$

Spring constant formula for springs in parallel?

$$k_eq = k_1 + k_2 + … + k_n$$

Spring constant formula for springs in series?

$$\frac{1}{k_eq} = \frac{1}{k_1} + \frac{1}{k_2} + … + \frac{1}{k_n}$$

What does having springs in parallel do to a system?

makes an equivalent spring that is stiffer

What does having springs in series do to a system?

Makes an equivalent spring that is weaker

Formula for force of gravity?

$$F_g = \frac{GM_1M_2}{R²}$$

Universal gravitational acceleration formula?

$$g = \frac{GM}{R²}$$

What is Newton’s Shell Theorem?

Assume a mass is in a shell casing. The shell casing’s Fg can be ignored since the forces in all directions are equal but opposite, leading to them all cancelling out.

What is the force of gravity inside an object?

$$F_g = \frac{GM_{inside}m}{R²}$$

Formula for resistive force?

$R = -bv$ or $R = -bv²$

Sometimes b is replaced by k

Formula for terminal velocity?

$$v_T = \frac{mg}{b}$$

Formula for v at any time during freefall with air resistance?

$$v = \frac{mg}{b}(1-e^{\frac{-bt}{m}}) = v_T(1-e^{\frac{t}{\tau}})$$

What does $\tau$ equal in freefalls and resistive forces?

$\tau = \frac{m}{b}$ and $\tau$ is the time constant

Acceleration at any point during freefall with resistive forces?

$$a = ge^{\frac{-bt}{m}}$$

Drag equation?

$$F_D = \frac{1}{2}D\rho Av²$$

D = drag coefficient

$\rho$ = density of air

A = cross-sectional area

V = velocity of object

Sum of forces in circular motion?

$$\sum \overrightarrow{F} = ma_c = m\frac{v²}{r}$$

What force causes the circular motion of an object moving on the ground?

friction force

Sum of forces for conical motion?

$$\sum F_y = 0 = Tcos(\theta) - mg$$

$$\sum F_x = Tsin(\theta) = ma_c$$

Velocity of an object in conical motion?

$$v = \sqrt{Lgsin(\theta)tan(\theta)}$$

Velocity of an object in conical motion using tension force?

$$v = \sqrt{\frac{Tr}{m}}$$

Max speed an object can negotiate a curve?

$$v = \sqrt{\mu_s gr}$$

Angle of a banked curve?

$$\theta = arctan(\frac{v²}{rg})$$

Sum of forces in a car in banked curves?

$$\sum \overrightarrow{F} = m\frac{v²}{r}$$

Tension for any point of an object moving in a vertical circle?

$$T = mg(\frac{v²}{rg} + cos(\theta))$$

Sum of forces at the top of a vertical circle?

$$ma_c = T_{top} + mg$$

Sum of forces at the bottom of a vertical circle?

$$ma_c = T_{bot} - mg$$

Sum of forces at any point in a vertical circle?

$$ma_c = T’ - mgcos(\theta)$$

Tension at the top of a vertical circle?

$$T_{top} = mg(\frac{v²}{Rg}-1)$$

Tension at the bottom of a vertical circle?

$$T_{bot} = mg(\frac{v²}{Rg}+1)$$

Velocity at the top of a vertical circle?

$$v_{top} = \sqrt{Rg}$$

Work formula for a force at an angle?

$$W = Fdcos(\theta)$$

Work formula (dot product)?

$$W = Fd$$

Do perpendicular forces do work?

no

Is work done with 0 displacement?

no

Work for a non-constant force?

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

Kinetic energy work equation?

$$W = \Delta K = \frac{1}{2}mv_f² - \frac{1}{2}mv_i²$$

Potential energy for springs?

$$U_s = \frac{1}{2}k\Delta x²$$

Work done by springs?

$$W_s = \frac{1}{2}kx_f² - \frac{1}{2}kx_i²$$

What are conservative forces?

Forces where the work done is not dependent on its path

What are nonconservative forces?

Forces where the work done is dependent on the path

Change in potential energy formula?

$$\Delta U = -\int_{a}^{b} F_{cons}(x) d\overrightarrow{x}$$

Conservative force to potential energy equation?

$$F_{cons} = -\frac{dU}{dx}$$

Work and potential energy equation?

$$W = -\Delta U$$

What is the work done in a closed path by a conservative force?

0J

What is the mechanical energy of a system?

E = K + U

What is stable equilibrium?

a minima on the potential energy graph

What is unstable equilibrium?

a maxima on the potential energy graph

Mechanical energy for systems with only conservative forces?

$$E = K_F + U_F = K_i + U_i$$

Change in kinetic energy for systems with friction?

$$\Delta K = \sum W - f_kd$$

Mechanical energy for systems with nonconservative forces?

$$U_i + K_i + W_{noncons} = U_2 + K_2$$

What are the formula(s) for power?

$P = \frac{dE}{dt}$ or $P = Fv$ (dot product)

Formula for average power?

$$P = \frac{W}{\Delta t}$$

Kinetic energy formula (using momentum)

$$K = \frac{p²}{2m}$$

Momentum formula (using kinetic energy)

$$p = \sqrt{2Km}$$

Force equation (using momentum)

$$\sum F = \frac{dp}{dt}$$

Impulse formula(s)?

$$I = \Delta p = p_f - p_i = \int_{t_1}^{t_2}Fdt$$

Force of thrust equation?

$$F = M\frac{dv}{dt} = |v_e\frac{dM}{dt}|$$

Sum of forces in rocket propulsion?

$$\sum F = -v_e\frac{dM}{dt}$$

Change in velocity for rocket thrust?

$$\Delta v=v_eln(\frac{M_i}{M_f})$$

Center of mass formula(s)?

$$x_{cm} = \frac{\sum m_ix_i}{\sum m_i}$$

$$y_{cm} =\frac{\sum m_iy_i}{\sum m_i}$$

Integral form for center of mass?

$$r_{cm} = \frac{1}{M}\int rdm$$

Linear density formula?

$$\lambda = \frac{M}{L}$$

Arc length formula?

$$S = r\theta$$

Angular formula for linear velocity?

$$v = r\omega$$

Angular formula for linear acceleration?

$$a = r\alpha$$

Average angular velocity formula?

$$\omega_{avg} = \frac{\theta_f-\theta_i}{\Delta t}$$

Average angular acceleration formula?

$$\alpha_{avg} = \frac{\omega_f-\omega_i}{\Delta t}$$

What are the four angular kinematics equations?

$$\omega_f = \omega_i + \alpha t$$

$$\theta_f = \theta_i + \omega_it + \frac{1}{2}\alpha t²$$

$$\omega_f² = \omega_i² + 2(\alpha)(\Delta \theta)$$

$$\theta_f = \theta_i + \frac{1}{2}(\omega_f + \omega_i)t$$

Total acceleration formula?

$$a = a_t + a_c$$

Magnitude of total acceleration formula?

$$||a|| = \sqrt{||a_t||²+||a_c||²} = r\sqrt{\alpha²+\omega^4}$$

Dot product for torque?

$$\tau = rFsin(\phi) = Fd$$

Net torque formula?

$$\sum \tau = I\alpha$$

Moment arm formula?

$$d = rsin(\phi)$$