a) Frequency is the number of complete oscillations per second. Looking at the graph, there are 9 complete oscillations in 5 seconds. Thus,
Frequency = 9/5 = 1.8 oscillations per second
Frequency = 1.8 Hz
Period = 1/frequency = 1/1.8
Period = 0.056 s
b) When we differenctiate displacement with respect to time, the result is velocity.
Recall, period = 1/f = 5/9 cycles
1/4 cycle behind = 1/4 x 5/9 = 5/36
It is delayed with 5/36 sec with respect to displacement.
5/36 sec = 0.139 sec
Acceleration = first derivative of velocity = second derivative of displacement = 1/4 cycle behind velocity = 1/2 cycle behind displacement =
5/36 = 0.139 sec delayed with respect to velocity
= 5/18 = 0.2777 secs delayed with respect to displacement
Thus, the number of seconds out of phase with the displacements is 0.278 seconds
c) The formula for calculating the period of an ideal pendulum anywhere is
T = 2π√length/local gravity). We would calculate the local gravity.
From the information given,
length = 0.2
T = P = 5/9
Thus,
5/9 = 2π√0.2/local gravity)
(5/9)/2π = √0.2/local gravity
Square both sides. It becomes
[(5/9)/2π]^2 = 0.2/local gravity
local gravity = 0.2/[(5/9)/2π]^2
local gravity = 25.56 m/s^2
Thus,
acceleration due to gravity = 25.56 m/s^2
Recall, earth's gravity = 9.8 m/s^2
number of g forces = 25.56/9.8
number of g forces = 2.61
(with negative sign since it is due south)
, so the average acceleration of the ball is given by