Simple Harmonic motion, circular motion and oscillations Flashcards
Uniform circular motion is when
an object rotates at a steady speed
f =
1/T
V =
2piR/T
The angular displacement of an object is given by
2pitf = 2pit/T
t = time T = period of one rotation
The angular speed, w =
angular displacement per second therefore
w = 2pi/T = 2fpi
The velocity of an object moving around a circle at constant speed continually…
changes direction towards the centre of the circle therefore accelerates towards the centre centripetal acceleration
The velocity of an object in uniform circular motion at any point is…
along the tangent to the circle at that point
acceleration =
v^2 / r
Centripetal force is
the resultant force on an object moving round a circle at constant speed
F =
mv^2/r = mw^2
theta =
2pi * f * t
w =
v/r = theta/t = 2pi*f = angular speed (rad/s)
a =
v^2 / r = w^2r
T =
1/f
sinX =
X for small angles
w =
2pi / t
Acceleration is always in opposite directions to the
displacement
Simple harmonic motion =
oscillating motion when acceleration is proportional to the displacement in the opposite direction
a (SHM) =
-kx = -(2pi f)^2x = -w^2x
Displacement of bob at time t is x =
Acos(2pi * ft)
The resultant force acting towards the equilibrium position is called the
restoring force
To decrease the frequency of the spring you can
add extra mass
use weaker springs
a (springs) =
-km/x
T = (spring) =
1/f = 2pi * (m/k)^0.5 for angles less than 10 degrees
T = (string) =
2pi * (L/g)^0.5 = mv^2 / L + mg
Potential energy =
0,5 * spring constant * x^2 where x = distance/amplitude
Energy changes between
kinetic and gravitational
A freely oscillating object =
no frictional forces so constant amiplitue
Dissipative forces are
The oscillations of a simple pendulum gradually die away because air resistance decreases the total energy of the system. The forces causing this decrease of amplitude are called dissipative forces because they dissipate the energy of the system to the surroundings as thermal energy.
If dissipative forces then the motion is said to be
`damped
amplitude =
maximum displacement from equilibrium
Period =
one complete cycle
Frequency =
number of cycles per second (Hz)
Light damping =
Time period is independent of amplitude so each cycle takes same length of time as oscillations die away
Amplitude decreases reducing by the same fraction each cycle
Critical damping =
just enough to stop the system oscillating after it has been displaced and released from equilibrium.
Heavy damping =
strong damping means the displaced object returns to equilibrium much more slowly than if the system is critically damped. No oscillating motion occurs
All types of harmonic oscillators include
variations of kinetic and potential energy
Energy of oscillator is proportional to
a^2
motion in circular path at constant speed but varying velocity implying
an acceleration and therefore a force
Rod in circular motion, when is tension max and min
Max: bottom of circle, T = F+W
Min: top of circle, T = F-W
Tension > weight because
circular motion has net force inwards so tension = F+/- W
Damping graph, energy graph, displacement time graphs
a
The simple pendulum is an example of a system that
oscillates with simple harmonic motion
w for pendulum =
(g/l)^0.5
Period of pendulum, T =
2pi / w = 2pi (L/g)^0.5
To test for simple harmonic motion check for
a restoring force that acts directly proportion to the displacement which is increasing
Mass-spring system, F = , a =, w^2 =, T =
F = -kx a = -kx/m w^2 = k/m T = 2pi (m/k)^0.5
When is tension max or min
The tension will be largest at the lowest point because it has to then supply the centripetal force and overcome the weight of he particle. It will be smallest at the highest point since here the weight is contributing to the centripetal force.
For simple harmonic motion, a =
-kx = (-2pif)^2 * x = -w^2x
= -(2pif)^2(A)(cos2pift) = -w^2 * Acos(wt)
If starts at max displacement
For simple harmonic motion, x =
Acos(2pi*ft) = Acos(wt) provided it starts at maximum displacement
T =
1/f = 2pi / w
For simple harmonic motion, x =
Asin(2pi *ft) = Asin(wt) provided it starts at equilibrium
Vmax =
amax =
2pifA = wA (maximum then all energy is potential
(2pi*f)^2A = w^2A
For simple harmonic motion, v =
+- 2pif(A^2 - x^2)^0.5
Total energy is directly proportional to
the amplitude squared
Graph of how potential energy varies with displacement
y = x^2
Graph of how kinetic energy varies with displacement
y = -x^2
For a pendulum, T=
2pi*(L/g)^0.5
For a mass-spring system, T =
f =
2pi*(m/k)^0.5
1/2pi * (k/m)^0.5
Damping force is directly proportional to the
negative of velocity (opposite direction)
