A bath toy is floating stationary in water such that half of the toy is above water and half of the toy is below the water. A kid pulls the toy 3.9 cm down into the water and then lets go. The toy then proceeds to bob up and down. It bobs up and down 26 times in a minute.
NOTE: Let up be positive and down be negative for this problem
(a) What is the Period of the motion?
(b) What is the frequency of the motion?
(c) What is the Amplitude of the motion?
(d) What is the equation that describes the position of the object at any given time?
period = T = 60 seconds/26 = 30/13
f = 1/T = 13/30
A = 3.9 cm
y = -3.9 cos (2 pi t/T)
because y = -3.9 when t = 0
y = - 3.9 cos (26 pi t/30)
To answer these questions, we need to understand the concepts of period, frequency, amplitude, and the equation describing the position of an object in simple harmonic motion.
(a) The period (T) of the motion is the time it takes for the toy to complete one full up-and-down cycle.
To find the period, we need to determine the time it takes for the toy to bob up and down once. Since we are given that it bobs up and down 26 times in a minute, we can calculate the period as:
T = 1 / (number of bobs per minute)
T = 1 / 26 bobs/minute
(b) The frequency (f) of the motion is the number of complete cycles (up and down) the toy makes in a given period of time.
The frequency is the reciprocal of the period:
f = 1 / T
(c) The amplitude (A) of the motion is the maximum distance the toy moves from its equilibrium position. In this case, it is the distance the toy is pulled down into the water.
Given that the toy is pulled 3.9 cm down into the water, the amplitude is 3.9 cm.
(d) The equation that describes the position of the object at any given time is called the displacement equation. It relates the position of the object (y) to time (t) and involves the amplitude (A) and angular frequency (ω).
For simple harmonic motion, the displacement equation is given by:
y = A * sin(ω * t)
where:
- y is the vertical displacement of the toy from its equilibrium position (positive when above, negative when below)
- A is the amplitude of the motion
- ω (omega) is the angular frequency, equal to 2πf (where f is the frequency)
In this case, we have the amplitude (A) and we can calculate ω using the known frequency (f).
ω = 2πf
Once we have the angular frequency, we can use the displacement equation to describe the position of the object at any given time.