[PHYSICS 1A03] - Final Exam Guide - Comprehensive Notes fot the exam (26 pages long!)

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PHYSICS 1A03 Full Course Notes

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Whenever we state a position, it is relative to some reference point or origin. When discussing the motion of an object over many successive positions during a time interval, we refer to this as trajectory (it is written as x(t), which emphasizes that the position x is a function of time. Displacement of an object is the change in position. Remember, distance is a scalar, while displacement is a vector. Distance also measures the total length of the trajectory. It doesn"t matter how you got from start to nish, displacement only depends on the starting and ending positions. We de ne velocity as the rate of change of an object"s displacement. The slope of a position vs. time graph is velocity. The sign of the velocity vector is the same as the displacement vector, meaning that velocity has direction. Speed is the rate of change of an object"s distance.

Related Questions

Galileo thought about whether acceleration should be defined as the rate of change of velocity over time or as the rate of change in velocity over distance. He chose the former, so let’s use the name “vroomosity” for the rate of change of velocity over distance. For the motion of a particle on a straight line with constant acceleration, the equation v = v_i + at gives its velocity v as a function of time. Similarly, for a particle’s linear motion with constant vroomosity k, the equation v = v_i + kx gives the velocity as a function of the position x if the particle’s speed is v_i at x = 0. (a) Find the law describing the total force acting on this object of mass m. Describe an example of such a motion or explain why it is unrealistic for (b) the possibility of k positive and (c) the possibility of k negative.

violethorse873

Learning Goal: Understand how to determine the constants in thegeneral equation for simple harmonic motion, in terms of giveninitial conditions.
A common problem in physics is to match the particular initialconditions - generally given as an initial position x_0 andvelocity v_) at t=0 - once you have obtained the general solution.You have dealt with this problem in kinematics, where theformula

1. x(t) = x_0 + v_0t + at^2

has two arbitrary constants (technically constants of integrationthat arise when finding the position given that the acceleration isa constant). The constants in this case are the initial positionand velocity, so "fitting" the general solution to the initialconditions is very simple.

For simple harmonic motion, it is more difficult to fit the initialconditions, which we take to be

x_0, the position of the oscillator at t = 0, and
v_0, the velocity of the oscillator at t = 0.

There are two common forms for the general solution for theposition of a harmonic oscillator as a function of time t:

2. x(t) = A\cos(omega t + phi) and
3. x(t) = C\cos(omega t) + S\sin(omega t),

where A, phi, C, and S are constants, omega is the oscillationfrequency, and t is time.

Although both expressions have two arbitrary constants--parametersthat can be adjusted to fit the solution to the initialconditions--Equation 3 is much easier to use to accommodate x_0 andv_0. (Equation 2 would be appropriate if the initial conditionswere specified as the total energy and the time of the first zerocrossing, for example.)

Find C and S in terms of the initial position and velocity of theoscillator.
Give your answers in terms of x_0, v_0, and omega. Separate youranswers with a comma.

[PHYSICS 1A03] - Final Exam Guide - Comprehensive Notes fot the exam (26 pages long!)

PHYSICS 1A03 Full Course Notes

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