Physics C: Mechanics

General Notes

1. What to remember with Dot vs Cross Product

2. Component notation

3. What to remember with any vector quantities?

4. What does r̂ mean?

5. How should every solution include?

6. What are the four approaches to a problem?

1. dot is scalar and uses cos, cross is vector and uses sin.

2. î for x-axis, j for y-axis, k for z-axis (convention is that into the page is negative)

3. Pythagorean theorem the components, remember SIGN

4. Unit vector for distance from an origin or axis of reference

5. Units, a form of calculus.

6. Momentum, Energy, Forces, mechanics/kinematics

Kinematics

1. When can kinematic equations be used?

2. x(t), v(t), a(t)

3. Kinematics graphs

4. Displacement

5. Projectile motion (where is v_y = 0, time)

1. ONLY if a is constant

2. MUST label as x’(t), x’’(t)

3. Be careful if its time vs position or some other quantities

4. Use the integral, remember bounds and initial condition

5. v_y=0 at max, v_x always constant, half the time to go up (if same distance back down)

Forces and Translational Dynamics 1

1. Newton’s 1st Law

2. Newton’s 2nd Law (and its calculus form

3. Newton’s 3rd Law

4. How to calculate COM and velocity for nonlinear mass

5. Acceleration of center of mass

6. 4 Types of forces that typically act on a stationary mass

7. Friction

8. Ideal Pulleys/Atwood Machines

1. An object in motion stays in motion (dynamic or static equilibrium)

2. ΣF = ma, can be used for single object or system. F = dpdt or chain rule version

3. Action-reaction, equal in magnitude opposite on direction, acting on two DIFFERENT objects

   • x_cm = ∫ λxdx / ∫ λdx, where dm = λdx

4. velocity is same thing but with v instead of x

5. a_cm only accelerates if NET EXTERNAL Force

6. F_g, F_T, F_N, F_f

7. F_sf < µF_N , Static vs Kinetic, µ is greater than static. Force of static friction matches force until force overcomes and then µ changes to kinetic

8. Tension is same, can ignore mass/friction, so its just simple sum of forces

Forces and Translational Dynamics 2

1. Incline Planes (and how to find frictional force)

2. Uniform circular motion (types, and energy approach to it)

3. Kepler 3 Laws* (don’t have to memorize)

4. Newton Universal Gravitation (using period)

5. Newton’s Law for Extended Bodies

6. How to find gravitational force if inside the planet

7. How elliptical orbits affect planets

1. Use tilted coordinate system, split F_g into F_parallel and F_perp to find normal force, accelerating force given the angle.

   • friction force is equal to mgsinØ given that no net force. µ = tanØ

2. constant radius, mv²/r, net force and acceleration towards center. Could be multiple forces contributing to F_c since its not a real force, like friction, gravity, normal force, etfc

   • No net work because force is perpendicular to motion

3. First Law: every planet moves in elliptical orbit. There are two foci of the ellipse, one of which being the sun. Second Law: The change in position between two points in the orbit happen under the same amount of time if the area of the positions from the sun is the same (see p. 188 of review book). Third Law: If T is period and a is length of semimajor axis of orbit (from center to farthest outside) then T²/a³ is the same for all planets orbiting the same star.

4. Big G formula, can also be modeled centripetally, DONT have to include the radius of the planet, only the distance between them. Remember that v = 2πr/T

5. HAVE to include the radius of the object for the formula because the distance between is not substantially larger than the object itself.

6. Must use M = ρV = ρ4/3πr³ and integrate or you can just use the ratio for M_within/M which equals Mx³/R³

7. Closer to planet means higher KE, less PE, tradeoff.

Work, Energy, & Power

1. Conservation

2. Types of U

3. Work

4. How to find PE

5. Power

6. Gravitational PE for planets

7. Escape velocity

1. K+ U (all types) + W before =after

2. U_g, U_s, U_e, U_M, U_internal

3. W = ∆K = ∫Fdr, remember the cosØ and ±

4. Use conservation with K, F = -dU/dx

5. P = dW/dt = F • v

6. U_g = –GM₁M₂/r

7. Velocity needed such that the KE is equal to or greater than the U_g between the planet and object

Linear Momentum

1. What is needed to change momentum?

2. Impulse-Momentum Theorem

3. Conversation

4. Collision Types

5. What to remember on momentum with components/directions

1. A force

2. J = ∆p = m∆v = ∫Fdt

3. Conserved only if closed system. Momentum is always conserved.

4. In elastic K is conserved. Completely inelastic means objects stick together.

5. REMEMBER DIRECTIONS, separate into components

Torque & Rotational Kinematics

1. How to convert linear/translation qualities to angular

2. What Rotations are measured in and what is the convention

3. Torque and directions

4. How Rotational Inertia works and how the formula differs for shapes

5. Nonlinear rotational inerta

6. Parallel-axis theorem

7. Types of Equilibriums

8. Rolling vs slipping

9. How does gravity act as a torque?

1. Multiply by r

2. θ, measured in radians, S=θr. 2π = 1 rotation. Counterclockwise is positive.

3. τ = F x r, remember sinθ. Use hand rule and remember directions

4. Farther the mass is distrubted from center, the more rotational inertia, ∫r²dm, mr² for hoop/point mass, decreases if closer to the center, like cylinder is 1/2mr²

5. Use dm = ρdA or dV depending on situation

6. I = I_cm+mx² where x is distance from regular axis and m is the total mass. Note that this only works for constant density

7. Translational vs Rotational vs Static Equilibrium

8. Rolling means both rotational and translational motion, slipping means only translational motion

9. Acts from COM

Energy and Momentum of Rotating Systems

1. Rotational Conservation

2. Which moves faster, ball slipping or rolling?

3. Rotational Work and Power

4. Angular Momentum and its Conservation

1. Remember tradeoff between K_translation + K_rotational as well as PE

2. Slipping bc no KE wasted on rotation. Also there’s always friction if rolling so probably slower

3. W = ∫τdθ, P = τω

4. L = pr = mvr = Iω (vector). τ = dL/dt, remember that L can transfer into p

Oscillations & Harmonic Motion

1. Springs/Hooke’s Law (spring constants for multiple springs)

2. Differences in energy at each position

3. Pendulum (forces and assumptions)

4. Physical Pendulum (and differences)

5. Period of pendulum vs Spring

6. Differential equation for harmonic motion

1. F=-kx, SHM, restoring force, Spring is Hookian if force is proportional to compression or stretch

   • Springs in series are Σ1/k, parallel is k1+k2

2. Highest KE/v at equilibrium point, highest PE/F/a at most ∆x.

3. Fg, remember the restoring force is only the HORIZONTAL component of Fg. Use soh cah toa with the angle. Can model the angle sinuoidally. If angle is small <15 then sinθ ~ θ

4. Pendulum period only depends on length and gravity while spring only depends on spring constant and mass

5. An object that has a COM that is a distance d away from axis (can be treated like point mass). Use a torque approach where -dMgsinθ = Id²θ/dt². Generally less torque/inertia bc mass distributed closer to the axis. T = 2π√(I/mgr)

6. F=-bx = ma = md²x/dt² = -bx, 2nd order or use the angle for a pendulum given that s=θr

Fluids

1. What to plug in for pVg

2. What can be said about pressure in a fluid/hydraulics?

3. What causes buoyant force?

4. How to find pressure?

5. What can be said about mass flow rate?

6. A slower fluid does what to the pipe?

7. The pressure out a spout is equal to what?

8. What happens if you’re calculating the magnitude of a force from inside the object?

9. what is the only thing that impacts if a block sides down a ramp?

10. What to remember when doing work?

11. How should you do conservation of momentum?]

12. How to find work for rotational motion?

13. If doing waves or sinusoidal curves, what should you remember?

1. p is density of fluid, V is volume of displaced fluid (object)

2. Pressure increases with depth, pressure is uniform throughout a fluid so F/A = F/A

3. Buoyant force is net upward force due to pressure being greater on the bottom of the object than top

4. P = P_gauge+P_atm

5. Mass flow rate is always equal in a pipe no matter its volume or size. pAv = pAV

6. put MORE pressure

7. Same as if gravity would be pushing it out. Using conservation of energy for fluids.

8. You must take into account density and volume

9. coefficient of friction in reference to angle of ramp (tan)

10. Sign of work is very important, depends on angle between force and distance

11. Note that system does not change unless there is a force then do components of momentum vector (check is angular momentum b/c its conserved)

12. ∫Torque dtheta

13. Which frequency is being used, radians

Additional Notes/Questions

1. How to find dy/dx

2. where does gravity act?

3. What does negative force or PE mean?

4. when converting from angular to linear quantities, what do the linear quantities represent about the motion?

5. What is true if rolling AND slipping

6. How does Fg act on a system?

7. Form of institutional force

8. Hand rule for torque

9. What does more uniform massdistrubtion mean for rotations?

10. What to remember about nonideal pulleys

11. what does drag depend on?

12. How can linear and rotational acceleration differ on a disc

13. what does a physical pendulum differ in?

14. where does normal force act on a horizontally hanging shelf?

1. use angle and tantheta (calc)

2. at the center

3. Negative force is attractive, negative PE means there’s less energy than at infinitely far away (needs to do work to separate)

4. TANGENTIAL component

5. omega does NOT equal r *v_tangential bc there is both rotaitoinal and translational v and KE. Rolling without slipping means FRICTION IS PRESENT acting as the centripetal force

6. Centripetal, use period plugin for Kepler’s third law

7. second derivative equals -constant*quantity

8. Curl fingers in direction of rotation, thumb is torque. OR: middle is torque, force is index, radius is thumb

9. less I, more omega

10. two different tensions, pulley has its own normal force (f_axle) and Fg

11. Its acceleration depends on both velocity and mass (differential equation)

12. all points of a disk may have same rotational acceleration but b/c diff radius, diff linear a

13. You have to use a torque approach with the center of mass (use linear density), also resistutional force is in terms of theta. Use rotational intertia and parallel axis theory

14. acts to the right.

YOU CANNOT JUST USE THE CENTER OF MASS TO FIND ROTATIONAL INERTIA

gravity acts at COM NOT geometric center, so sometimes it does cause torque

conservation of torque, newtons laws (3rd) apply to toruqe too

dropping a mass onto a rotating disk is a collision

Small angle approximation for pendulums