This force causes the circular motion to change direction. It is directed towards the centre.

_____F = ma = _____as v = rw_______F = mw²r

Rounding a bend

mg - R = _____ ___ => tan q =

Bucket of water

A bucket of water is swung in verticle circle

______ mg < Þ water stays in bucket.

Pendulum

A pendulum made to move in a horizontal circle, with tension T in the string.

___ ___Tsinq = ___Tcosq = mg ___Þ tanq =

From prolonged observations of planets Kepler produced 3 laws

1. Earth moves in an ellipse which has the sun as one focus.
2. The line joining the sun to a moving planet sweeps out equal areas in equal times.
3. r³ µ T², _r - radius of planet-sun, T - period of planet around sun.

"Every particle of matter in the universe attracts every other particle with a force which is directly proportional to the product of their masses and inversely proportional to the square of their distance apart. "

Action at a distance, effect of a body A exerting a force on another body B. Due to a gravitational field around A.

Gravitational field strength, ____g = = _______

___Variation of g with height.

_______

___

When Fc = Fg \ no net force experienced in ship, no net acceleration relative to ship

__________________________

For a body in a lift descending___ mg - S = ma___ if S = mg Þ weightlessness ( S is normal reaction )

In oscillatory motion the acceleration, displacement & velocities change periodically (in magnitude & direction).

Definition of SHM - If the acceleration of a body is directly proportional to its distance from a fixed point and is always directed towards that point, the motion is simple harmonic.

__________________________

Time peiod , T - time to complete one cycle or oscillation.
Frequency, f - number of oscillations per second.
Amplitude, A or r - maximum displacement from mean position.
Phase, f - the initial angular state of the system, usually as a fraction of the time period.
Phase difference, Df degree to which oscillations are out of step, in relation to the time interval between maximum displacements._______Df = (time difference at maximas) / (time period)

acceleration,_a µ - x,_______a = - k x = - w x² = - w²A sin wt,____ velocity,_v = - v_m cos wt = w ± ( A² - x² )^½
max. velocity, v_m = ± wA____displacement,_x = A sin wt = A sin ( wt + f )__if x = 0 at t = 0 then f = 0,
(q = wt, v_m = wA ), __T =

_______________________________

Hooke's law :________T = ke, \ T₀ = mg = ke₀ ( equilibrium )
When displaced, x : ___F = mg - T = mg - k ( e₀ + x ) = mg - mg - kx
Newton's 2^nd law : ___.F = ma, \ ma = - kx \ a = \ period T = 2p

SHM Experiment to find 'g' using mass on a spring
_____
From (4.11): T² = 4p² and

for x & q small ® sin q =
Force towards centre ( to tension ), F = - mg _____ T =
SHM Experiment to find 'g' using a pendulum

F = mg ____ acc. = - wx² = g ____ Þ w² = ____\ w =
As T = ____then T² =

If g = 9.8 ms^{- 2} then the experiment has accurately been measured for SHM where acc. = - wx² .

PE = ½ F x____( F = k x )____\ PE = ½ k x² = ½ m w²y² ____(amplitude a² = x² + y², y can be incorporated into diagram 4.9)
KE = ½ m v² = ½ m w²x²____E = KE + PE__ (conservation of mechanical energy)

Damping :

The graph relates to the experiment with 'Bartons pendulum' & also other SHM systems.
____
Examples of resonance Þ air column in musical instrument, radio tuned to natural frequency = radio signal (electrical resonance), Tacoma Narrows Bridge disaster - wind caused oscillating force in resonance with natural frequency of the bridge.

Experiment to find resonant amplitude - Barton's pendulums

Frequency, f, found from time exposure photographs. Resonant pendulum = 1/4 oscillation behind diver pendulum.

1. If s = r then q = 1 radian ( rad ) - 1 rad is the angle subtended at the centre of a circle by an arc = in length to the radius. The circumference is s = 2pr where q = 2p rad. ( or 360° )

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