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Calculus
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Collected Resources
Calculus Overview
Velocity Dependent Forces
Simple Harmonic Motion Oscillators
Rotational Inertia and SHM Period Bypasses
Mechanics C Calculus Summary
x
(m)
Derivative
x
t (s)

v
(m/s)

Derivative
 vdt
 adt
v
dx
dt
 Fdt
Derivative
t (s)
F
(N)
dp
F
dt

t (s)
Derivative
Integral 
F
(N)
W
 Fdx
x (m)
Integral
F
dU
dx
Integral
t (s)
dv
a
dt
t (s)

J
U
or
W
(J)
Integral
v

a
(m/s2)
J
or p
(N.s)
x (m)
Notice
 how the units can helpyou remember
this, derivatives are slopes, divide the Y units by
the X units, integrals are like area, multiply the
Y units by the X units, replace x with θ, v with
ω, and a with α, for rotational kinematics
Velocity Dependent Force problem
Car coasts to a stop, find v(t)
Vo
Assume FD = kv
N
+y
FD
+x
Fx  ma
FD  ma
dv
kv  m
dt
dv k

v
dt m
v  Ae Bt  C
mg
Guess a
dv
function
 BAe Bt
for v that dt
is equal
k
Bt
Bt
BAe

(Ae
 C)
to its own
m
derivative

t (s)
dv
k
 g v
dt
m
v  Ae
v  v 0 at t  0
v  Ae 0 , A  v 0
v  v 0e
Works for freefall with air
resistance too:
v
(m/s)
k
t
m
k Bt k
BAe 
Ae 
C
m
m
Bt
k
C  0, B 
m
Yields:

k
t
m
mg
v
(1 e
k
k
t
m
)
Simple Harmonic Motion
Fs = kx
Solve for x(t)
Fs
+y
+x
mg
F  ma
2
d x
dt 2
Guess x  Acos(Bt)  C
dx
 BA sin( Bt)
dt
d2x
2

B
Acos(Bt)
2
dt
d 2 x k


x
dt 2 m
k
B 2 Acos(Bt) 
(Acos(Bt)  C)
m
kx  m
N
k
(Acos(Bt)  C)
m
k
k
B 2 Acos(Bt) 
Acos(Bt) 
C
m
m
k
k
C  0 ,  B2 
, B
m
m
B 2 Acos(Bt) 
k
x  Acos(
t)
m
at t  0 , x  x M ,  
2
k
m
k
x  x M cos(t) ,  
,   2

m

Calculus Bypasses:
m
Find the Period of a
Torsion Pendulum
l
Find I about the end of a thin rod
IT  ICM  mh

(N. m)
Slope = k
2
l 2
1 2
IT  ml  m 
2 
12
1 2
l2
IT  ml  m
12
4
1 2 3 2
IT  ml  ml
12
12
4 2 1 2
IT  ml  ml
12
3
θ (r)
Slope is the Force
(Torque) Constant
Works for I
about any
point on the
rod
Inertia
  2
Force Constant
  2
I
k