F=ma class 6
bobyhaha
July 2022
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Introduction
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Oscillation
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In general, the oscillations have the form of mef f ddt2x +kef f x = 0, and the general
q
k f
solution is y(x) = Asin(wt) + Bcos(wt) = Ccos(wt + ϕ), where w = mef
.
ef f
The value of A and ϕ can be determined by the initial condition.
For an object, the total torque isτ = Σri × mi g = Σmi ri × g = M rc × g, and
thus we can say the total net torque is equivalent as acting on the center of
mass.
dw
If an object makes a small angle ∆θ with the vertical axis, since τ = dL
dt = I dt =
d2 θ
I dt2 , and we define the distance between the center of mass of the object with
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the pivot point as l, we have −mglθ = I ddt2θ .
q
This looks highly similar to the above formula so we can conclude w = mgl
I
When the problem asks for period, we use the formula T = 2π
w
In general, to solve this type of problems, we make a small displacement dx, dθ
from the equilibrium position and see the small change in dF, dτ , then we just
plug in the formula and find the oscillation period/frequency.
There is a different way to solve this kind of problem, which is to use the energy
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2
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and we can get things in the form 12 mef f vef
f + 2 kef f xef f = E and take the
derivative with respect to time to solve this kind of problem. Now let’s solve a
lot of problems to get ourselves familiarize with oscillation
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Torque
In general if the system is under equilibrium, we can write out the equations
ΣFi = 0
Στi = 0
We can calculate the total torque using the equation τ = r × F
However, there are other ways to solve an equilibrium problem, which is to use
purely geometry, and in most case, it involves triangle.
Angle of friction is also really important, where as if an object is just on the
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f
⃗
verge of sliding, N
= µs . In most case, we can combine f⃗, and N into T⃗ = f⃗+ N
Examples will be shown in class.
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Fluid
Using energy conservation we can have the Bernoulli’s equation. Prove will
shown in class. NOTE: This can only be used if the velocity at specific time is
not changing with time
From infinitesimal method, we can see that p(h) = p0 + ρgh.
NOTE: This equation can only be applied when the whole system is STATIC.
We also can have gef f to deal with pressure when we are at non-inertial reference
frame, and that gef
⃗ f = ⃗g − ⃗a
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Using power to solve problems
⃗ v , and in some cases it is an useful tool to
Power is defined to be P = dE
dt = F · ⃗
solve problems( shown in class)
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Mathematical tools
• Series
• Taylor Expansion
• Average
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Adiabatic Theorem
Σone period pdx = constant This can be used to solve collision problems.
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