**Mechanics**

March 05 2018

**Motion**
- law of inertia
- inertia is the tendency of objects to continue in its state of rest or uniform motion in a straight line
- mass is a measure of inertia
- objects will undergo changes in motion only in the presence of a net force
- mechanical equilibrium:
- for all objects at rest, the net forces acting on it are zero
- for all objects moving at uniform motion, the net forces acting on it are zero
- acceleration is measured as the rate of change of velocity
- average a = change in v / change in t
- average v = change in s / change in t
- instant a = dv/dt
- instant v = ds/dt
- v = S a dt
- s = S v dt
- s = 1/2 a t2 for initial zero v, initial zero a, and constant a
- 1/2 because of the difference between average and instant velocity
- as expressed by act of integration
- ramp race is won by track with faster average velocity due to earlier acceleration
- law of acceleration
- when a net force acts on an object, it will accelerate
- the acceleration is directly proportional to the net force and inversely proportional to the mass
- a = net force / m
- force, velocity, and acceleration are all are vector quantities with both magnitude and direction
- a car driving in a circular track is accelerating because the direction of its velocity is changing
- acceleration is a vector in the same direction as the net force
- applied at right angles, the force will deflect the object
- any other direction will result in a combination of speed change and deflection
- weight = gravitational force
- mass = amount of matter inducing inertia to resist the change
- a boulder 100 times more massive than a pebble free-falls with the same acceleration
- because although the force on the boulder (its weight) is 100 times greater
- its resistance to a change in motion (its mass) is 100 times greater
- the greater mass (inertia) offsets the greater weight (force)
- for gravitational free-fall
- this results in the same acceleration ratio
- because gravitational force is proportional to mass
- law of action-reaction
- whenever one object exerts a force on a second object
- the second object exerts an equal and opposite force on the first
- a net force external to the system is required to move the system
**Energy**
- energy is a fundamental aspect of the universe
- the combination of matter and energy make up the universe
- energy cannot be created or destroyed, it can only change form
- conservation of energy
- in the absence of an external force
- the total energy of a system remains unchanged
- one way to transfer energy to a system
- is to act on it with an external force
- we say the force does work on the system
- work
- work = external constant force x displacement in the direction of the displacement
- work = | force | x | displacement | x cos angle
- this combination occurs so often that it is given a special name
- the scalar product or dot product of two vectors
- work = external force dot displacement
- the unit is a newton-meter or joule
- both force and displacement are vectors, work is a scalar quantity
- no work is done by a perpendicular force, negative work is done by a force acting in the opposite direction of the displacement
- more generally
- work is the line integral of a varying force over an arbitrary path
- W = S F dot dr
- kinetic energy
- doing work on a system by applying a force
- is the mechanical way to transfer energy to the system
- how does that energy manifest itself
- under some conditions it shows up as kinetic energy
- energy of the system's motion
- work-kinetic energy theorem in one-dimension:
- W net = S F dx = 1/2 mv2 2 - 1/2 mv1 2
- K = 1/2mv2
- change in K = W net
- like velocity, kinetic energy is a relative term
- its value depends on the reference frame
- unlike velocity, kinetic energy is a scalar and is never negative
- power
- climbing a flight of stairs requires the same amount of work no matter how fast you go
- but it's harder to run than to walk
- harder in the sense that you do the same work in less time
- power is the rate of doing work
- P average = change in work / change in time
- P instant = dW / dt = F dot v
- the unit is joules / second or watt
- conservative force
- where the total work by by a force acting as an object moves over any closed path is zero
- where the general formula for work is the line integral W = S F dot dr
- S over closed path F dot dr = Sc F dot dr = 0
- since the return path between two points must be the reverse amount of work
- the work done by a conservative force in moving between two points is independent of the path taken
- SAB F dot dr depends only on endpoints A and B
- potential energy
- the energy stored by virtue of an object's position that has the potential to be converted into kinetic energy
- the amount of energy is the amount of work required to acquire the position in countering a conservative force
- change in U = - SAB F dot dr
- in the one-dimensional case where the force and displacement are parallel and when the force is constant
- change in U = - F (x2 - x1)
- change in gravitational potential energy = weight x height = mgh
- conservation of mechanical energy
- the work-kinetic energy theorem shows that
- change in kinetic energy = Wnet
- in cases where the only forces acting are conservative
- change in potential energy = - Wnet, so
- change in kinetic energy + change in potential energy = 0
- or kinetic energy + potential energy = a constant
- the total mechanical energy of a system does not change
**Potential Energy Curves**
- graphs of potential energy by position
- knowing the total energy of the system
- allow us to find the turning points that determine the range of motion available to the system
**Gravity**
- a weak universal force of attraction
- F = G m1 m2 / r2
- where G is a constant
- gravitational field
- provides a way to describe gravity that avoids the troublesome action-at-a-distance
- a gravitating mass creates a field in the space around it
- a second mass responds to the field in its immediate vicinity
**Systems of Particles**
- center of mass
- vector r cm = sum mi ri / M or = S r dm / M
- we can then apply Newton's second law to the entire system of particles
- a complex system acts as though all its mass were concentrated at the center of mass
- net external force = M acm = dP / dt
- where acm and P are the acceleration of the center of mass and the momentum of the system
- this works because internal forces cancel out
- conservation of linear momentum
- in the absence of an external force
- the momentum of a system remains unchanged
- when net external force = 0, P = constant
- amazingly, this applies even in the subatomic realm where Newtonian laws fail
- collisions can also be used to study the mass of subatomic particles
- without understanding the forces that are acting
**Rotational Motion**
- angular acceleration
- angle in radians is the dimensionless measure s / r
- the circumference is 2 pi r, so one revolution is 2 pi
- angular velocity is revolutions per second, or radians per second
- angular velocity = w = d angle / dt
- it is analogous to linear velocity
- since angle = s/r
- we can relate linear velocity to angular velocity
- d angle / dt = 1/r ds/dt, so v = wr
- angular acceleration = ang = dw/dt
- tangential acceleration at = dv/dt = r dw/dt = r ang
- whether or not there is angular acceleration
- points on a rotating object also have radial acceleration because they are in circular motion
- ar = v2/r = w2r
- the total acceleration of the points on a rotating object
- are the vector additions of tangential and radial acceleration
- a = at + ar
- torque
- it would be cumbersome to apply Newton's second law to all particles of a rotating object
- instead we find analogues of force and mass
- just like how angular acceleration is the analogue of linear acceleration
- torque is the effectiveness of a force in bringing about change in rotational motion
- it depends on the perpendicular component of the force and distance from the rotating axis, or equivalently, on the force and the effective distance, the lever arm
- t = r F sin angle = r F perpendicular = r perpendicular F
- the units for torque, newton-meters, are the same as for energy
- but it is a different quantity than energy
- so we reserve the term joule (= 1 Nm) for energy
- mass
- it is easier to set an object rotating when its mass is concentrated near the rotation axis
- so angular inertia depends on both the mass and its distribution relative to the rotation axis
- Newton's second law
- F = m at = m r ang
- since torque = r F if the force is applied at right angles
- torque = mr2 ang, or t = I ang
- this is the rotational analog of F = ma
- it can be extended to rigid bodies
- I = sum mi ri 2 = S r2 dm
- energy
- rotational kinetic energy is the sum of the kinetic energies of all the parts
- rotational kinetic energy = 1/2 I w2
- work = change in rotational kinetic energy
- vectors
- angular velocity as a vector points in the direction depicted by the right-hand rule
- for magnitude-only changes in angular velocity, angular acceleration points in the same direction
- torque is proportional to angular acceleration, and should point in the same direction
- torque = r F sin angle
- the right-hand rule rolling from radius to force points in the direction of the torque
- the direction is perpendicular to both the vectors of r and F
- this operation occurs frequently and is called the cross product
- C = A x B
- is a vector whose magnitude is A B sin angle and whose angle is perpendicular to both A and B
- torque = r x F
- angular momentum
- the momentum form of Newton's Second Law was very useful
- the same applies to angular momentum
- L = r x p
- torque = dL / dt
- the rotational analogue for momentum of Newton's Second Law
- conservation of angular momentum
- in the absence of an external torque
- angular momentum is constant
- the angular momentum of a system is conserved
- because a composite system can change its configuration
- and hence its rotational inertia I
- conservation of angular momentum requires angular speed increase if I decreases
- precession
- the conservation of angular momentum does not specify how or about what axis something has to rotate
- as long as the system's total angular momentum is conserved
**Static Equilibrium**
- equilibrium is when the net external force and torque are both zero
- static equilibrium is when the body is also at rest
- selecting a pivot point can help simplify the torque equilibrium calculations
- the center of gravity is the point at which the gravitational force seems to act
- the center of gravity coincides with the center of mass when the gravitational field is uniform
- equilibrium states include stable, unstable, neutrally stable, and conditionally stable