physics 1b03summer - lecture 7 homework question please do this question and hand it by tuesday...
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Physics 1B03summer - Lecture 7
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
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
Oscillatory Motion
Physics 1B03summer - Lecture 7
Spring and mass
MEquilibrium: no net force
M
The spring force is always directed back towards equilibrium. This leads to an oscillation of the block about the equilibrium position.
M
For an ideal spring, the force is proportional to displacement. For this particular force behaviour, the oscillation is simple harmonic motion.
x
F = -kx
Physics 1B03summer - Lecture 7
)cos( tAxSHM: x(t)
t
A
-A
T
A = amplitude
= phase constant
= angular frequency
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
Three constants specify the motion:Amplitude, AAngular Frequency, Initial phase (or phase constant),
)cos()( tAtxIn general:
t
x(t)
Φ
These graphs are a mathematical representation of motion as a functionof time, now how the object actually moves – notice the axes. x(t) is simply the displacement from some position.
Physics 1B03summer - Lecture 7
)360(or radians 2 if
)0()( so
)2cos( cos(0)
T
xTx
AAx
2 2
fT
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:
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).
M
x
v0
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:
M
x0
v0
A) 0o
B) 30o
C) 60o
D) -30o
E) -60o