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