Transcript Potential Energy - McMaster University

HOMEWORK QUESTION
Please do this question and hand it by Tuesday after
the reading week, in class:
A 50kg child slides down a 45o frictionless hill for 60m,
starting with an initial velocity of 2m/s. The child then
slides for 10m over a flat surface that has a
coefficient of kinetic friction of 0.15, and finally back
up another frictionless hill with a slope of 30o.
Draw a pictures of the problem and determine how far
on the 2nd hill the child ends up (not the height).
Physics 1B03summer - Lecture 7
Oscillatory Motion (Chapter 14)
• Kinematics of Simple Harmonic Motion
• Mass on a spring
• Energy
Knight sections 14.1-14.6
Physics 1B03summer - Lecture 7
Oscillatory Motion
We have examined the kinematics of linear motion with
uniform acceleration. There are other simple types of
motion.
Many phenomena are repetitive or oscillatory.
Example: Block and spring, pendulum, vibrations
(musical instruments, molecules)
M
Physics 1B03summer - Lecture 7
Spring and mass
M
Equilibrium: no net force
The spring force is always directed back
towards equilibrium. This leads to an
oscillation of the block about the
equilibrium position.
M
F = -kx
M
For an ideal spring, the force is
proportional to displacement. For this
particular force behaviour, the oscillation
is simple harmonic motion.
x
Physics 1B03summer - Lecture 7
SHM:
x  A cos(t   )
x(t)
A
T
A = amplitude
t
 = phase constant
 = angular frequency
-A
A is the maximum value of x (x ranges from +A to -A).
 gives the initial position at t=0: x(0) = A cos .
 is related to the period T and the frequency f = 1/T
T (period) is the time for one complete cycle (seconds).
Frequency f (cycles per second or hertz, Hz) is the number of
complete cycles per unit time.
Physics 1B03summer - Lecture 7
In general:
x(t )  A cos(t   )
x(t)
Φ
t
Three constants specify the motion:
Amplitude, A
Angular Frequency, 
Initial phase (or phase constant), 
These graphs are a mathematical representation of motion as a function
of time, now how the object actually moves – notice the axes. x(t) is
simply the displacement from some position.
Physics 1B03summer - Lecture 7
The quantity ( t + ) is called the phase, and is measured in radians.
The cosine function traces out one complete cycle when the phase
changes by 2 radians. The phase is not a physical angle!
The period T of the motion is the time needed to repeat the cycle:
x(0)  A cos  A cos(2   )
so x(T )  x(0)
if  T  2 radians (or 360)
2

 2f
T
units: radians/second or s-1
Physics 1B03summer - Lecture 7
Example
The block is at its equilibrium position and
is set in motion by hitting it (and giving it
an initial velocity) at time t = 0. Its motion
is SHM with amplitude 5 cm and period 2
seconds. Write the function x(t).
v0
M
x
Physics 1B03summer - Lecture 7
QUIZ
The block is at x0 = +5 cm, with positive
velocity v0, at time t = 0. Its motion is SHM
with amplitude 10 cm and period 2
seconds. If x(t) = A cos (t  ), the phase
constant  should be:
A)
B)
C)
D)
E)
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v0
x0
0o
30o
60o
-30o
-60o
Physics 1B03summer - Lecture 7