Final cram review

*Conversions:

1000g = 1kg
kg —> N = kg*9.8
100cm = 1m

Unit 1: Kinematics

Vectors = magnitude and direction
Scalars = magnitude only
Displacement is vector, distance is scalar
Velocity = vector
Acceleration = vector (v/t), if very small considered instantaneous (focus on acceleration at a certain point in time instead of over an amount of time)
Slope of position vs time graph = velocity
Slope of velocity vs time graph = acceleration - area between curve and horizontal axis of graph is change in position
Area between curve and horizontal axis of accel. vs time graph is change in velocity
Area above axis is positive, area below is negative
USE 10 FOR GRAVITY NOT 9.8
Projectile motion (If near Earth surface) - y = 10m/s², x = 0 (v is constant)

Unit 2: Force and Translational Dynamics

Center of Mass - x = mx/m (total), x can be replaced with v or a
All forces are vectors and a result of interaction between two objects
Tension force is always parallel to direction of rope/pull
Newton’s First Law = An object at rest will remain at rest unless acted upon by an external force, Law of Inertia = tendency of an object to resist acceleration
Newton’s Second Law - a = F/m
Translational equilibrium = object is at rest or constant velocity (a = 0)
Newton’s Third Law = Every force exerted by object one to object two, object two exerts a force equal and opposite to that of object one (simultaneously)
Kinetic friction = Two surfaces sliding relative to one another (Friction = coefficient of friction * normal force)
Static friction = Two surfaces not sliding relative to one another, this frictions prevents them from doing so (Friction is less than or equal to coefficient of friction * normal force)
Force of friction does not depends on size of surface area of contact between two surfaces
Force of gravity = m*g = G*m of plant (* m of object)/r of planet^2
An ideal spring force is proportional to its displacement from equilibrium position (Equation is Hooke’s law)
Direction of tangential (circular motion) velocity is always changing
Centripetal force not in FBD, in direction is positive and out direction is negative

Unit 3: Work, Energy, and Power

Translational Kinetic Energy = object whose center of mass is moving
Work = amount of mechanical energy transferred into or out of system,
Work done by a conservative force on a system is independent of the path of the object (gravitational and spring force)
Work done by a nonconservative force on a system does depend on the path of the object (force of friction and air resistance)
Potential energy - Kinetic, gravitational, and elastic
Potential energy is stored in a system due to position of the object in system
A single object cannot have potential energy
General form (always negative) of gravitational potential energy exists between 2 objects with mass (0 Ug = 2 objects infinitely far away from one another)
Work and energy are scalars (Work, KE, Ug, and Ue {elastic} do not have direction, only magnitude)
Conservation of Energy - a system with only one object can only have kinetic energy
Total Mechanical Energy stays the same if 0 net work is done or 0 work done by nonconservative forces
When net work is done on a system energy is transferred between the system and its surroundings
When work is done by friction and NO external force adds or removes energy, the work done by a nonconservative force = change in mechanical energy
Work-Energy Principle = W done by nonconservative force = change in KE, Always valid even when friction and external force acts on system
Identify initial, final, and horizontal zero line of system
Power = the rate which energy changes in certain amount of time, either transferred in or out of a system or converted from one type of energy to another within the system

Unit 4: Linear Momentum

Linear momentum = vector
External net force applied to a system is negligible relative to internal forces acting on the system
During an explosion, forces internal to the system push objects in the system apart
Impulse (J) = change in momentum or F*change in time (vector), area under force vs time graph curve
When net force on a system = 0, impulse = 0, no change in momentum (none added or removed)
If net force acting on a system with multiple objects is 0, momentum = 0, velocity is constant
When net force is not 0, impulse is not 0, momentum is transferred between system and environment
Elastic collisions - KE before = KE after, Kinetic energy is conserved
Inelastic collisions - KE before is more than KE after, KE decreases
Perfectly inelastic collisions - objects stick together, KE decreases
Most real world collisions are inelastic, momentum is ALWAYS conserved, KE is conserved only in elastic

Unit 5: Torque and Rotational Dynamics

1 revolution = 360 degrees = 2pi radians
Angular velocity = angular displacement/change in time
Angular acceleration = angular velocity/change in time
Rigid objects with shape maintain a constant shape as they rotate
All points in a rigid object go through the same angular displacement during the same time, have the same angular velocity and acceleration
An object moving along a circular path moves through angular displacement
Also moves through a linear distance - arc length = radius*angular displacement
Rotational variables need to be in radians (dimensionless quantity)
Centripetal acceleration = tangential velocity²/radius = radius*angular velocity²
Three different types of accelerations an object can have in circular motion: Angular acceleration, tangential acceleration, and centripetal acceleration
Angular acceleration is an angular quantity measured in units of rad/sec²
Tan. acceleration is tangent to a circular path being traced by an object and perpendicular to the radius
Tangential acceleration refers to change in magnitude of the tangential velocity of an object
Centripetal acceleration is directed perpendicular to the circular path being traced by an object and in towards the center of the circle along the radius
Centripetal acceleration refers to the change in direction of the tangential velocity of an object
Circular motion cannot exist without centripetal acceleration
Tan. and Cen. acceleration are always perpendicular to each other
Tangential and centripetal acceleration are measured in units of m/s²
When a rigid object is rotating, angular displacement, tangential velocity, and tangential acceleration refer to the whole object, while arc length, tangential velocity, tangential acceleration, and centripetal acceleration refer to the specific location on the object
Torque = the ability of a force to cause an angular acceleration of an object
r sub perpendicular = lever arm = perpendicular distance from the axis of rotation to the line of action of the force
Torque is a vector (equation on sheet is magnitude of it)
Rotational Inertia = measure of how much an object resists angular acceleration
Parallel axis theorem (used to identify axis of rotation on an object parallel to another) = I = I (center of mass) + mass*distance²
When a system is in rotational equilibrium, the angular velocity of the system is constant
Newton’s First Law Rotational = An object at rest must remain at rest, a rotating object maintains a constant angular velocity, unless acted upon by an external torque or the distribution of the mass of the object changes
Newton’s Second Law Rotational = torque = Inertia*angular acceleration
Object at rest and not rotating is in both translational and rotational equilibrium = static equilibrium
In static equilibrium the net torque about any axis of rotation is equal to zero (you can pick axis of rotation)

Unit 6: Energy and Momentum of Rotating System

Rotational Kinetic Energy - objects whose center of mass is changing location have translational kinetic energy, objects that are rotating have rotational kinetic energy (1/2*I*angular velocity²)
Total Kinetic energy of rigid object = KE rotation + KE translational
A torque can transfer energy into or out of a rigid system if the torque acts over an angular displacement
Work done by a constant torque on a rigid system = W = torque*angular displacement
On a graph of torque as a function of angular displacement, work is the area under the curve
Angular momentum of a rigid object with shape has to be relative to an axis of rotation
Angular momentum is a vector
Angular momentum of a point particle - L = r*m*v*sin, finds magnitude of the angular momentum of a point particle
Angular momentum of a point particle has to be relative to an axis of rotation of reference line
Angular Impulse - torque = change in angular momentum/change in time, derive to find angular momentum = torque*time = area under torque vs. time curve
On a graph of angular momentum as a function of time, the slope of the curve is the net torque acting on the object
Conservation of angular momentum = total angular momentum of a system remains constant if the net torque acting on the system equal zero, no angular momentum is added or removed from the system
Rolling without slipping for rigid object with shape:
Displacement = r*angular displacement
Velocity = r*angular velocity
Acceleration = r*angular acceleration
Objects rolling without slipping have both translational and rotational kinetic energies
Acceleration of rigid object with shape rolling without slipping on incline depends only on three variables - angle of incline, gravitational force, and factor in front of mr² in rotation inertia equations (ex. acceleration of a solid cylinder rolling without slipping down an incline - a = 2/3*g*sin)
Object rolling with slipping invalidates equations for rolling
Motion of Orbiting Satellites:
Circular orbits - system - Total mechanical energy = constant & Total gravitational potential energy = constant, satellite - Angular momentum = constant & Kinetic energy = constant
Elliptical orbits - system - Total Mechanical energy = constant & Total gravitational potential energy = not constant, satellite - Angular momentum = constant & Kinetic Energy = not constant
For satellite in either orbit - linear momentum is not constant due to change in direction of velocity, angular momentum is constant due to direction of angular velocity remaining constant
Escape velocity is the speed at which an object must travel to break free from a planet or moon's gravitational force and enter orbit. It can calculated using: $v_e = \sqrt{\frac{2GM}{r}}$ , where $G$ is the gravitational constant, $M$ is the mass of the planet, and $r$ is the distance

Unit 7: Oscillations

Periodic motion = motion which is repeated in equal intervals of time
Simple Harmonic Motion = periodic motion which results from a restoring force acting on an object where magnitude of that force is proportional to displacement of object from equilibrium position (location where net force acting on object is zero)
Restoring force is always directed towards equilibrium position
Period of SHM = time it takes to go through one full cycle or oscillation
Amplitude of SHM = maximum distance from equilibrium position
Mass spring system:
Position 1 (stretched) - max distance stretched (amplitude), velocity is 0, spring force and acceleration are at max and directed to the left
Position 2 (equilibrium) - displacement is 0, velocity is at max (either directed left or right), spring force and acceleration are 0
Position 3 (compressed) - max magnitude in the negative direction (negative of the amplitude), velocity is 0, spring force and acceleration are again at max and now directed to the right
Horizontal mass-spring system restoring force is spring force acting on the mass
Simple pendulum is considered to be in SHM for small angles (max angle can be as large at 15 degrees), restoring force is component of gravity acting on the pendulum tangent to direction of motion of the pen.
Position of object in SHM = x = Acos(2pi*f*t)
cos (initial position of the object is A) vs. sin (initial position of the object is equilibrium position) = shifted from one another by a magnitude of 90 degrees or pi/2 radians
Total mechanical energy = KE + PE
In a Isolated system the TME is constant
Horizontal, ideal mass-spring system: ME = 1/2kA², to find max speed - sqrt.kA²/m = A*sqrt.k/m
Unit 8: Fluids
Pressure is scalar
Buoyant force is equal in magnitude to the weight of the fluid displaced by the object
Ideal fluid flow in a closed volume (ex. pipe) - volumetric flow rate (Av) = constant
When A decreases, v increases
Bernoulli’s equation is a description of mechanical energy remaining constant in ideal fluid flow
Bernoulli’s principle relates fluid speed and fluid pressure, assuming difference in height is negligible, according to his principle, if fluid speed increases, fluid pressure decreases
Torricelli’s theorem: speed of an ideal fluid exiting a large, open reservoir through a small hole: v2 = sqrt.2*g*h1

Example to find tension of a cable:

Calculate the buoyant force: The buoyant force is equal to the weight of the air displaced by the balloon. Use the formula: $F{buoyant} = V{balloon} * \rho{air} * g$ where $V{balloon} = 120 m^3$ , $\rho_{air} = 1.2 kg/m^3$ , and $g = 9.8 m/s^2$ (although using 10 m/s² is recommended in the notes for ease of calculation).

Calculate the weight of the gas inside the balloon: Use the formula: $W{gas} = V{balloon} * \rho{gas} * g$ where $V{balloon} = 120 m^3$ , $\rho_{gas} = 0.19 kg/m^3$ , and $g = 9.8 m/s^2$ (or 10 m/s²).
Calculate the weight of the balloon material: This is given as 50 kg. So, $W{balloon} = m{balloon} * g = 50 kg * 9.8 m/s^2$ (or 10 m/s²).
Calculate the tension in the cable: The tension in the cable is the difference between the buoyant force and the total weight of the balloon and the gas inside it. Use the formula: $T = F{buoyant} - W{gas} - W_{balloon}$

Now, let's plug in the values and calculate:

Buoyant Force:
$F_{buoyant} = 120 m^3 * 1.2 kg/m^3 * 10 m/s^2 = 1440 N$
Weight of Gas:
$W_{gas} = 120 m^3 * 0.19 kg/m^3 * 10 m/s^2 = 228 N$
Weight of Balloon:
$W_{balloon} = 50 kg * 10 m/s^2 = 500 N$
Tension in the Cable:
$T = 1440 N - 228 N - 500 N = 712 N$

So, the tension in the cable is approximately 712 N.