** indicates a detail required of only University Physics students (phy 231 and phy 241)
* indicates a detail required of only College and University Physics students (not required of phy 121 students)
|
1.1. Definition of Average Rate: Average rate of change of A with respect to B = change in A / change in B.
1.2. Primary example of a rate of change: average velocity, in miles / hr is average rate of change of position with respect to clock time: Divide miles traveled (change in position) by hours required (change in clock time) and you get average velocity in miles / hr.
1.3. Average rate of change of y with respect to x in terms of graph: Ave rate = change in y / change in x = rise / run = slope.
1.4. Change in y corresponding to time interval: Change in y = ave rate of change of y with respect to t * `dt (abbreviated: `dy = ave rate * `dt)
Idea 2: Velocity and Acceleration as Rates
2.1. Velocity is rate of change of position with respect to clock time.
2.2. Acceleration is rate of change of velocity with respect to clock time.
2.3. If acceleration is uniform on an interval, then the graph of v vs. t is linear on that interval, and vice versa
2.4. Uniformly accelerated motion is conceptualized in terms of the seven quantities displacement, time interval, ave velocity, init velocity, final velocity, change in velocity and acceleration, symbolized `ds, `dt, vAve, v0, vf, `dv, and a. We need to understand all these quantities and how they are related to one another in order to understand uniformly accelerated motion.
Idea 3: Applying the Ideas of Rates to Position, Velocity and Acceleration
3.1. On a graph of position vs. clock time
3.2. On a graph of velocity vs. clock time
3.3. On a graph of acceleration vs. clock time
4.1. For uniform acceleration the four equations of uniformly accelerated motion apply:
4.2. Given any of the three quantities `ds, `dt, v0, vf and a we can use these equations to find the other two.
4.3. Uniformly accelerated motion can therefore be analyzed, if not completely understood, in terms of the five quantities `ds, `dt, v0, vf and a.
4.4. The quantities vAve and `dv can easily be determined if we know `ds, `dt, v0 and vf. While vAve and `dv are essential to understanding uniformly accelerated motion, they are not necessary to analyze uniformly accelerated motion.
Idea 5: Sequential Observations and Graphical Representation of Positions and Velocities
5.1. From position vs. clock time data we can calculate vAve = `ds / `dt for each interval, giving us approximate v vs. t information.
5.2. From v vs. t data we can calculate aAve = `dv / `dt for each interval, giving us approximate a vs. t information.
5.3. From v vs. t data we can calculate vAve for each interval, from which we can calculate `ds = vAve * `dt for each interval. If we know initial position we can then add changes in position to get position vs. t information.
Idea 6: Newton's First Law and situations involving Uniform Acceleration
6.1. Acceleration in the direction of motion changes the speed of an object.
6.2. Acceleration perpendicular to the direction of motion changes the direction of motion but not the speed of an object.
6.3. Newton's First Law: If net force on it is zero an object will not accelerate.
An object which does not accelerate will change neither its speed nor its direction of motion.6.4. The vertical motion of an ideal projectile is characterized by the uniform acceleration of gravity, while its horizontal motion is characterized by zero acceleration. Horizontal and vertical motions are independent.
6.5. Acceleration on a frictionless incline with small slope is a = g * slope. This is an approximation that does not hold for large slopes.
Idea 7: Reasoning Out and Formulating Uniformly Accelerated Motion
For uniformly accelerated motion:
7.1. `ds = vAve * `dt gives us
7.2. `dv = a * `dt gives us
7.3. Eliminating vf, then eliminating `dt from these two equations gives us
Idea 8: Newton's Laws of Motion
8.1. Fnet = m a, where Fnet is the net force on mass m and a is the acceleration of the mass.
8.2. Standard situations include:
Idea 9: Work (net force applied through displacement) and the Work-Energy Theorem
9.1. `dW = Fave * `ds, where Fave is the average force in the direction of motion and `ds the displacement thru which the force acts.
9.2. KE = .5 m v^2, where KE is the kinetic energy of a mass m moving with velocity v
9.3. v = sqrt(2 KE / m), obtained by solving KE = .5 m v^2 for v, is the velocity of an object of mass m with kinetic energy KE
9.4. Fnet * `ds = `dKE, where Fnet is the average net force acting on the system in the direction of motion, `ds the displacement thru which the net force acts, and `dKE the change in the kinetic energy of the object. This is called the Work-Energy Theorem
9.5. `dWby + `dKE = 0, where `dWby is the work done by the system against the net force acting on the system and `dKE is the change in the KE of the system.
Idea 10: Impulse (force applied over time interval) and Momentum; Conservation of Momentum
10.1. p = m v, where p stands for momentum.
10.2. `dp = Fnet * `dt, called the Impulse-Momentum Theorem
10.3. pTotal1 = pTotal2, where pTotal1 and pTotal2 are the initial and final momentum of a closed system
10.4. m1 v1 + m2 v2 = m1 v1 ' + m2 v2 ' , where v1 and v2 are the velocities immediately before and v1 ' and v2 ' the velocities immediately after a collision between masses m1 and m2.
11.1. `dKE + `dWcons_by + `dWnoncons_by = 0, where `dKE is the change in the KE of a system, `dWcons_by the work done by the system against the net conservative force and `dWnoncons_by is the work done by the system against the total nonconservative force.
11.2. `dPE is the work done by the system against the net conservative force
11.3. `dKE + `dPE + `dWnoncons = 0 expressed the Work-Energy Theorem using `dPE instead of `dWnoncons.
For vectors in the x-y plane:
12.1. Where A is the magnitude of a vector and theta is the angle made by the vector with the positive x axis:
12.2. Where Rx and Ry are the components of the resultant vector R:
12.3. Vector quantities have magnitude and direction and include, among many others:
12.4. The following, among many others, have no direction and therefore not vector quantities:
Idea 13: Gravitation is an Inverse-Square Force
13.1. F = G m1 m2 / r^2, where G is the universal gravitational constant, m1 and m2 the masses of the two objects and r the distance between the centers of mass.
Idea 14: For objects in circular orbits, Centripetal Force = Gravitational Force
14.1. Fcent = v^2 / r, where Fcent is the centripetal force holding the object in its circular path, v the velocity of the object and r the radius of the circle
14.2. m v^2 / r = G M m / r^2 expresses the equality between centripetal force and gravitational force on a satellite with mass m in its orbit around a planet of mass M, with M >> m and r = radius of orbit
14.3. v = sqrt( G M / r ), the solution for v to the preceding equation.
Idea 15: Change in gravitational PE = Average gravitational force * Displacement
15.1. `dPE = Fave * `dr, where Fave is the average of the (nonlinear) gravitational force on an object and `dr the change in the distance of the object from the center of a planet.
** `dPE = integral( G M m / r^2 with respect to r, from r1 to r2), where M is the mass of a planet, m the mass being moved from distance r1 to distance r2 with respect to the center of the planet.
15.2. `dPE = G M m / r1 - G M m / r2 is the precise change in gravitational PE between r = r1 and r = r2.
Idea 16: Between two circular orbits, `dKE = -1/2 * `dPE
16.1. PE = -G M m / r, the potential energy with respect to r = infinity of masses M and m separated by distance r
16.2. KE = 1/2 G M m / r, the potential energy of mass m in circular orbit of radius r about mass M, provided m < < M.
Idea 17: Angular Motion is completely analogous to Linear Motion
17.1. theta = s / r, where s is arc distance along a circle of radius r and theta is the corresponding central angle
17.2. theta is the measure of angular position with standard unit radians, analogous to position x or s in linear motion with standard unit meters
17.3. omega is angular velocity, the rate of change with respect to clock time of angular position with standard unit radians / second, analogous to rate of change of position v with unit meters / sec in linear motion
17.4. alpha is angular acceleration, the rate of change with respect to clock time of angular velocity omega, with unit radians / second^2; analogous to acceleration a, rate of change of velocity v with unit meters / second^2 in linear motion
17.5. s = r * theta, where s is arc length along a circle of radius r and theta the corresponding angular displacement in radians
17.6. v = r * omega, where v is speed along a circle of radius r and omega the corresponding angular velocity in radians / sec
17.7. a = r * alpha, where a is speed along a circle of radius r and alpha the corresponding angular acceleration in radians / sec^2
17.8. all the reasoning used to analyze linear motion can be applied to angular motion, including the equations of uniformly accelerated motion
Idea 18: Newton's Second Law can be formulated in terms of Angular Motion and Moment of Inertia
18.1. tau = F * r, where tau is the torque exerted by force F applied at distance r from axis of rotation, F perpendicular to moment arm. Torque in angular dynamics is analogous to force in linear dynamics.
18.2. tau = F * r * sin(theta), where tau is the torque exerted by force F applied at distance r from axis of rotation, F making angle theta with moment arm. Torque in angular dynamics is analogous to force in linear dynamics.
18.3. I = m r^2, the moment of inertia I of a particle at distance r from axis of rotation. Moment of inertia is analogous to mass in linear dynamics; just as greater mass requires greater force for a given acceleration, the greater the moment of inertia the greater the torque required to achieve a given angular acceleration.
18.4. I = sum( m r^2), the moment of inertia I of a collection of particles of various masses at various distances from axis of rotation. Moment of inertia is analogous to mass in linear dynamics; just as greater mass requires greater force for a given acceleration, the greater the moment of inertia the greater the torque required to achieve a given angular acceleration.
18.5. The moment of inertia of a homogenous circular cylinder with mass M and radius R with axis of rotation through the axis of the cylinder is I = 1/2 M R^2
18.6. The moment of inertia of a uniform homogeneous rod with mass M and length L rotating about an axis through the center of the rod and perpendicular to the rod is I = 1/12 M L^2.
18.7 tauNet = I * alpha, where tauNet is the net torque required to achieve angular acceleration alpha on an object with moment of inertia I. This is directly analogous to and is a direct result of Newton's Second Law Fnet = m a.
Idea 19: The definitions of Work and KE can be reformulated in terms of Angular Quantites
19.1. `dW = tau * `d`theta, where `dW is the work done by a torque tau acting through angle `d`theta. This is analogous to `dW = F * `ds for linear motion.
** W = integral ( tau with respect to theta, from theta1 to theta2)
19.2. KE = .5 * I * omega^2, where KE is the kinetic energy of an object with moment of inertia I rotating about its axis with angular velocity omega. Analogous to KE = .5 m v^2 for linear motion.
Idea 20: Angular Impulse-Momentum gives rise to a new conservation law.
20.1. angular impulse = tau * `dt, where tau is torque and `dt is time interval. Analogous to impulse `dp = F * `dt.
20.2. angular momentum = I * omega, where I is moment of inertia and omega is angular velocity. Analogous to linear momentum m * v . Just as change in linear momentum is equal to impulse, change in angular momentum is equal to angular impulse.
20.3. Angular momentum is conserved in a closed system. Angular momentum is not the same as linear momentum.
Idea 21: Simple Harmonic Motion results from a Linear Restoring Force
21.1. The equation Fnet = - k x, where Fnet is net force and x is displacement from equilibrium and k is a constant called the force constant, defines a linear restoring force
21.2. omega = sqrt( k / m ), where omega is the angular frequency of the simple harmonic motion of a mass m subject to a linear restoring force with force constant k.
21.3. x(t) = A cos( omega * t ) is an equation for the position x of an object in simple harmonic motion with angular frequency omega and amplitude A.
21.4. v(t) = -omega * A * sin(omega * t) is the velocity function of an object whose position function is x(t) = A sin( omega * t).
** the velocity function is the derivative of the position function
21.5. a(t) = -omega^2 * A * cos( omega * t ) is the acceleration function of an object whose position function is x(t) = A sin( omega * t).
** the acceleration function is the derivative of the velocity function, the second derivative of the position function
* 21.6. x(t) = A cos( omega * t + theta0 ) is one form of the most general equation for the position x of an object in simple harmonic motion with angular frequency omega and amplitude A.
* 21.7. x(t) = A sin( omega * t + theta0 ) is another form of the most general equation for the position x of an object in simple harmonic motion with angular frequency omega and amplitude A.
* 21.8. v(t) = -omega * A * sin(omega * t + theta0) is the velocity function of an object whose position function is x(t) = A sin( omega * t + theta0 ).
** the velocity function is the derivative of the position function
* 21.9. a(t) = -omega^2 * A * cos( omega * t + theta0 ) is the acceleration function of an object whose position function is x(t) = A sin( omega * t + theta0 ).
** the acceleration function is the derivative of the velocity function, the second derivative of the position function
* 21.10. x(t) = A sin(omega * t) or y(t) = A sin(omega * t) or x(t) = A sin(omega * t + theta0) or y(t) = A sin(omega * t + theta0) are also valid functions for the position of an object in simple harmonic motion with angular frequency omega and amplitude A.
22.1. x(t) = A cos(omega * t) is the x coordinate of a point moving on a reference circle of radius A, centered at the origin, with motion at constant angular velocity omega starting at the positive x axis when t = 0
22.2. x(t) = A cos(omega * t + theta0) is the x coordinate of a point moving on a reference circle of radius A, centered at the origin, with motion at constant angular velocity omega starting at angular position theta0 when t = 0
22.3. y(t) = A sin(omega * t) is the y coordinate of a point moving on a reference circle of radius A, centered at the origin, with motion at constant angular velocity omega starting at the positive x axis when t = 0
Idea 23: Velocity and Acceleration in SHM follow the Reference Circle Model
23.1. v = omega * A is the speed of the reference-circle point, which moves on a circle of radius A with constant angular velocity omega
23.2. a = v^2 / A = omega^2 * A is the centripetal acceleration of the reference-circle point.
23.3. v_x (t) = -omega * A * sin(omega * t) is the x component of the velocity of the reference-circle point which starts at the positive x axis when t = 0.
23.4. a_x (t) = -omega^2 * A * cos(omega * t) is the x component of the acceleration of the reference-circle point which starts at the positive x axis when t = 0.
Idea 24: Energy Relationships in SHM are consistent with the Reference Circle Model
24.1. PE_max = .5 k A^2 is the PE at amplitude A of a simple harmonic oscillator with force constant k
24.2. total energy = .5 k A^2 is the total energy of a simple harmonic oscillator with force constant k and amplitude A
24.3. PE = .5 k x^2 is the PE at position x of a simple harmonic oscillator with force constant k
24.4. KE + PE = total energy for a simple harmonic oscillator at any position.
24.5. KE = total energy - PE = .5 kA^2 - .5 k x^2 for a simple harmonic oscillator at any position.
24.6. vMax = speed of the point on the reference circle modeling SHM; recall that this speed is omega * A
24.7. .5 k A^2 = KEmax, the KE of mass m at velocity vMax = omega * A = sqrt( k / m) * A.
Supplemental ideas:
25.1 The motion of a point moving on a circle of radius A at constant angular velocity omega can, at any instant, be characterized by the radial vector r from the origin to the point, the velocity vector v tangent to the circle and hence perpendicular to the radial vector, and the centripetal acceleration vector a_cent directed toward the center of the circle and hence perpendicular to the velocity vector and parallel to but in the direction opposite the radial vector. The magnitude of r is the radius A of the circle, the magnitude of v is the speed r * omega of the point and the magnitude of the centripetal acceleration a_cent is v^2 / r, where v is the speed of the point.
25.2. If the point represents a mass m, then to remain on the circle the mass must a net force F_net = m * a_cent. If the mass is to represent a satellite of mass m in a circular orbit about a planet of mass M, then F_net = G M m / r^2, where the radius r of the orbit is equal to the radius A of the circle. The radial vector r at an instant represents the position of the satellite, v represents its velocity and a_cent its acceleration toward the center of the planet.
25.3 In the context of SHM we regard the circle as a 'reference circle', and the point as the 'reference point'. The motion of the simple harmonic oscillator is a projection of the motion of the reference point. Specifically, if omega matches the angular frequency of the oscillator and the radius A of the circle matches the amplitude of the motion, the projections of the r, v and a_cent vectors on any line through the origin coincide the position, velocity and acceleration of a simple harmonic oscillator.
25.4. When the projection line is the x axis, the circular model yields the position function x(t) = A cos( omega * t + theta0) of 21.6. The x components of the v and r vectors are the associated velocity and acceleration functions given in 21.8 and 21.9.
25.5. When the projection line is the y axis, the associated position function is y(t) = A sin (omega * t + theta0). The y components of the r vector and the a_cent vector give us the associated velocity and acceleration functions.
Idea 26: In an elliptical orbit the total mechanical energy is constant
26.1. In the orbit of any ideal satellite about a planet the only force exerted is the conservative force of gravity, so that `dW_nc_on = 0 and PE + KE = constant.
26.2. PE decreases as a satellite in an elliptical orbit approaches the planet, so since KE + PE = constant it follows that KE increases. Therefore the satellite speeds up as it approaches the planet.
26.3. When a satellite is approaching a planet in an elliptical orbit the component of the gravitational force tangent to its path is in the direction of its motion, so that gravity does positive work on the satellite, thereby decreasing its gravitational PE and increasing its KE. The satellite therefore speeds up. When the satellite is moving away from the planet the component of the gravitational force tangent to its path is in the direction opposite its motion, so that gravity does negative work on the satellite, increasing its gravitational PE and decreasing its KE. The satellite therefore slows.