Question

question 4 (1 point)
a pebble is embedded in the tread of a rotating bicycle wheel of diameter 60 cm. if the wheel rotates at 4 revolutions per second, determine a relationship between the height, h, of the pebble above the ground, in centimetres, as a function of time t, in seconds.
o a) ( h = 30 sin left( \frac { 1 } { 4 } pi t
ight) + 30 )
o b) ( h = 30 sin ( 8 pi t ) + 30 )
o c) ( h = 60 sin left( \frac { 1 } { 4 } pi t
ight) )
o d) ( h = 60 sin ( 8 pi t ) )

Explanation:

Step1: Determine the amplitude and vertical shift

The diameter of the wheel is \(d = 60\) cm, so the radius \(r=30\) cm. The vertical shift \(D\) (mid - line) is the height of the center of the wheel, \(D = 30\) cm. The amplitude \(A\) (maximum deviation from the mid - line) is equal to the radius of the wheel, \(A = 30\) cm.

Step2: Calculate the angular frequency

The wheel rotates at \(f = 4\) revolutions per second. The formula for the angular frequency \(\omega\) is \(\omega=2\pi f\). Substituting \(f = 4\) into the formula, we get \(\omega=2\pi\times4=8\pi\) radians per second.

Step3: Write the general form of the sinusoidal function

The general form of a sinusoidal function for height is \(h(t)=A\sin(\omega t)+D\). Substituting \(A = 30\), \(\omega=8\pi\) and \(D = 30\) into the formula, we get \(h(t)=30\sin(8\pi t)+30\)

Answer:

B. \(h = 30\sin(8\pi t)+30\)