Minimums and Maximums
Maximum distance from the equilibrium point a spring or pendulum can travel (denoted by 'A')
Amplitude
a spring has a period of 4000 milliseconds. Find the angular velocity with units
4000 ms = 4 s
2pi/4 =pi/2
pi/2 rads/sec
the equation for potential energy of a pendulum
U = mgh
time it takes for one oscillation
period
dictates the initial position of the oscillation or wave at time t=0 (φ)
phase constant
Equation for max velocity
Aw
a spring has an amplitude of 4 m, a max velocity of 16 m/s, and a mass of 3 kg. Find the spring constant
vmax=Aw
16=4w
w=4
4=(k/m)1/2 --> 4 = (k/3)1/2
16=k/3 k = 48 N/m
the equation for the potential energy of an oscillating spring
KE=1/2kA2cos2(wt)
number of oscillations that occur per unit of time (typically seconds)
frequency
units of frequency
Hz, Hetz, s-1
Aw2
equation for max acceleration
A spring with an amplitude of 3 m and k of 200 N/m starts at its max displacement. What is the kinetic energy at the equilibrium point
Usp=1/2kA2cos2(wt)
Uinitial=1/2(200)(3)2cos2(0)
Uinitial=900 J
At EP, all potential is kinetic energy, so KE = 900 J
a pendulum starts from max displacement. graph the kinetic energy
this stupid website won't let me insert pictures. your graph should start at zero and oscillate, never going below zero
equation used for the period of a pendulum
T=2π√L/g
the force directed against displacement to bring body back to equilibrium
restoring force
what happens to acceleration AND velocity at the equilibrium point?
Acceleration is zero, velocity is max
Using Newton's Law and Hooke's Law, explain why a spring in SHM has the equation x(t)=cos(wt)
(ignore phase constant for now)
F=ma
-kx=ma=m(d2x/dt2)
(d2x/dt2)=k/m(-x)
need a function whose second derivative is the negation of itself with a constant in front
a(t) =-w2cos(wt)
x(t) = cos(wt)
starting from the equilibrium point create a graph comparing the kinetic energy and velocity of an oscillating object
pretend there is an awesome picture here. both start at max, reach zero at same time, when KE is at max when velocity is max and min
You are constructing a grandfather clock. Assuming you want the clock to tick each second, what are the possible lengths you could make the pendulum
option 1: 1 meter
T=2pi(l/g)1/2
T=2pi(.1)1/2
T= 2 seconds --> ticks every half period
option 2: 1/5 meter
T= 1 second --> ticks every period
This states that for small angular displacements, the sine of the angle is approximately equal to the angle itself
Small angle approximation
A spring initially has an amplitude of 2 m and a period of 4. Assuming everything else remains constant, how does the period of the spring change when the amplitude is doubled? How would it change if it were a pendulum?
It doesn't. Period is independent of amplitude.
Using torque and moment of inertia, explain why a pendulum in SMH has the equation θ(t)=cos(wt)
(ignore phase constant)
torque =-mglsin(θ)=Ia
-mglsin(θ)=ml2(d2θ/dt2)
(d2θ/dt2)=-g/l sin(θ) --> small angle approx.
(d2θ/dt2)=-g/lθ
need a function whose second derivative is the negation of itself with a constant in front
a=-Aw2cos(wt)
θ(t)=cos(wt)
When is potential energy the greatest?
Accepted answers: when acceleration is greatest, max displacement, amplitude
A pendulum with a period of 1 s on Earth, where the acceleration due to gravity is g, is taken to another planet, where its period is 2 s. What is the acceleration due to gravity on the other planet.
Tp=2pi(l/g)1/2
1=2pi(l/g)1/2
for period to double, gravity would have to decrease by a factor of 4.
g/4
how do you find the spring constant for springs in series AND parallel spring
series: 1/k1 + 1/k2 = 1/keff
parallel: k1 + k2 = keff