Ch 14: Periodic Motion Flashcards
A body that undergoes periodic motion always has _____.
a stable equilibrium postion
Why does a body moved away from its equilibrium position overshoot the equilibrium position when released?
It has picked up some kinetic energy
In SHM, it’s simplest to define _____ as the origin.
the equilibrium postion
Whenever a body is displaced from its equilibrium position, the spring force _____. This is called _____.
tends to restore it to the equilibrium position; a restoring force
Oscillation can occur only when there is _____.
a restoring force tending to return the system to equilibrium
In SHM, when the body is at O, the net force acting on it is _____.
zero
When the body is to the left/right of equilibrium, the net force and acceleration are to the _____.
right/left
amplitude, A
the maximum magnitude of displacement from equilibrium; always positive
What is one cycle (give an example)?
one complete round trip: for example, from A to -A and back to A, or from O to A back to O to -A back to O
period, T
the time for one cycle; always positive; measured in cycles per second
frequency, f
number of cycles per unit time; measured in Hz
angular frequency, ω?
rate of change of an angular quantitiy (not necessarily related to a rotational motion); 2πf; measured in rad/s
How is ω related to T?
ω= 2πf = 2π/T
When does the simplest kind of oscillation occur and what is this called?
when the restoring force is directly proportional to the displacement from equilibrium; this is called simple harmonic motion
Hooke’s Law
F = -kx
What is the acceleration of a body in SHM?
-(kx)/m
In SHM, acceleration and displacement always have _____.
opposite signs
T/F: Acceleration in SHM is constant.
Wicked, tricksy, FALSE!
SHM is the projection of _____.
uniform circular motion onto a diameter
phasor
a rotating vector
What is the value of the x-component of the phasor at time t?
x=Acosθ
How is ω related to k and m?
ω = √(k/m)
In simple harmonic motion, the period and frequency do not depend on _____.
amplitude
What is the equation for displacement in SHM?
x=Acos(ωt + Φ)
What are the max and min values of x in SHM?
-A and A
Changing m or k changes _____.
the period of oscillation
phase angle, Φ
tells at what point in the cycle the motion was at t=0
If Φ=0, then x_0=
A, and the body starts at its max positive displacement
If Φ=π, then x_0=
-A, and the body starts at is max negative displacement
How is v_0 related to ω, A, and Φ?
v_0 = -ωAsinΦ
What is the equation for Φ?
Φ = arctan( - v_0/ωx_0)
What is the equation for A?
√(x_0^2 +v_0^2/ω^2)
When the body has both an initial displacement and a nonzero initial velocity, the amplitude is ______.
NOT equal to the initial displacement
What is total mechanical energy equal to in SHM?
E = ½mv^2 + ½kx^2 = ½kA^2 = constant
The velocity of a body in SHM is _____.
not constant
At x = ± A, the energy is all ______.
potential
T/F: There are major differences between vertical SHM and horizontal SHM.
FALSIES
When a body is hanging from a vertical spring in equilibrium, _____ is equal to _____.
the spring’s upward vertical force; the body’s weight; kΔl = mg
What is the only difference between vertical and horizontal SHM?
In vertical SHM, the equilibrium position x=0 no longer corresponds to the point at which the spring is unstretched
Describe angular SHM in a mechanical watch.
The wheel has a moment of inertia about its axis and the coil spring exerts a torque proportional to the angular displacement from equilibrium
What is ω equal to in angular SHM?
ω=√(κ/I)
What is the equation for displacement in angular SHM?
θ=Θcos(ωt +Φ)
Is the system of two atoms fundamentally different from a mass attached to a horizontal spring?
NOPE NOPE NOPE
What is a simple pendulum?
A point mass suspended by a massless, unstretchable string
What is the restoring force for a simple pendulum and what is it provided by?
F= -mgsinθ ; gravity
What is ω for a simple pendulum?
√(g/L)
What is a physical pendulum?
Any pendulum that uses an extended body
In the equilibrium position, the center of gravity of a physical pendulum is _____.
directly below the pivot point
When a physical pendulum is displaced from equilibrium by θ, the weight causes a restoring torque equal to _____.
-mgdsinθ, where d is the length from the pivot point to the center of equilibrium
If θ is small, we can make the approximation that _____.
sin θ = θ in radians, so τ = -mgdθ
What is ω for a physical pendulum?
√(mgd/I)
damping
the decrease in amplitude caused by dissipative forces
What is the net force on a body going under damped oscillation?
ΣF= -kx - bv
What is the displacement for a body going under damped oscillation?
x = Ae^[-(b/2m)t] cost ( ω’t + Φ)
What is the angular frequency for a body going under damped oscillation?
√(k/m - b^2/4m^2)
critical damping
when angular frequency is equal to zero; b = 2√(km); the system no longer oscillates but returns to its equilibrium position
overdamping
b > 2√(km) ; no oscillation, system returns to equilibrium more slowly
underdamping
b < 2√(km) ; system oscillates with steadily decreasing amplitude
