Simple Harmonic Motion
Oscillators
Universal Gravitation
Equations to Remember
100
What is the general expression for the position of an oscillator in SHM?
x=x_maxcos(omegat+phi)
100
The period of a spring-oscillator depends on __________ and does not depend on __________.
Depends on: mass & spring constant
Does not depend on: amplitude
100
The force of gravity on a mass is inversely proportional to the _________________.
distance squared (to the center of the celestial mass)
100
What is the orbital velocity of a satellite in a circular orbit?
v=sqrt((GM)/R)
200
What two conditions must be met for a restoring force to keep an object in SHM?
1. The magnitude of the restoring force is directly proportional to displacement.
2. The direction of the restoring force is opposite to the direction of the displacement.
200
What condition must be true for us to apply the equation
T_p=2pisqrt(l/g)
to a pendulum?
Small angle approximation - the initial angle of the pendulum must be small.
200
Write an expression you could use to relate the apogee and perigee velocities of a satellite in an elliptical orbit.
Either
r_amv_a=r_p"mv_b
or
K_a+U_a=K_p+U_p
200
What is the differential equation for any oscillator in Simple Harmonic Motion?
(d^2x)/(dx^2)=-omega^2x
300
A rod of mass M and length R has a rotational inertia of
I=1/4MR^2
and is pivoted to rotate a distance of R/2 from its center. Derive the angular frequency of the rod for small-angle oscillations.
omega=sqrt((2g)/R)
300
Two springs with spring constants k1 and k2 are connected in parallel to a mass M as shown below.
Right an expression for the period of the spring.
T_s=2pisqrt(M/(k_1+k_2
300
How would you derive Kepler's 3rd Law for circular orbits (
T^2=(4pi^2)/(GM)r^3
)? (Which equations would you use?)
a_c=r^2omega, g=GM/r^2, and T=(2pi)/omega
300
Write at least two different forms of the virial theorem (including K, Ug, and E).
K=-1/2U_g , E=-K, E=1/2U_g