trig questions?

Question

hi. plz help me with these, extra credit for school and i desperately need them. Plz explain how to do them and give the direct answers as well. thanks in advance:

2) On a steam locomotive the piston that drives the train moves the length of the diameter of the wheel (3 feet). If the train is moving at 30 miles per hour, write the equation of the displacement of the piston as a function of time.

Amplitude=

Revolutions per minute=

Revolutions per second=

w/omega

Equation d=

What is the displacement of the piston after 4 seconds



4) A water wheel has a barrel attached to it. The wheel is pushed by a creek at a constant rate of 3 rpm. Express the height of the barrel as a function of time in seconds.

Answer 1

The piston will have a sinusoidal motion described by the equation

d = Asin(omega*t)

EDIT: This assumes the zero point of displacement is the mid-point of the piston.

A is the amplitude of half the motion, in this case 1.5 ft; the total displacement is the positive amplitude minus the negative amplitude and you are given that as equal to the wheel diameter of 3 ft.

The wheel moves along the track one circumference every revolution. The wheel circumference is 3π ft. 30 miles per hour is 30/60 = 0.5 miles per minute. The wheel will rotate 5280/(3π) times per mile, or 0.5*5280/(3π) revolutions per minute. Divide that result by 60 to get revolutions per second. Omega is radians per second, and one revolution has 2π radians. Therefore omega equals 2π*0.5*5280/(3π) = 5280/3 radians per second.

Use this value of omega and 1.5ft for A to get the equation for displacement, and plug in 4 seconds for t to get the displacement after 4 seconds.

The second problem is just like the first, except that the height never goes less than zero: it goes from zero to the diameter of the wheel. The height is then given by

h = (D/2)sin[omega*(t-π/2)] + D/2

(Check it out: at t = 0, the sin function is -1 so the height is zero; at t = π, the sin function is 1, so the height is D.)

Omega is 3 rpm, or 3*2π radians per minute, or (3*2π)/60 radians per second.

You could also express this as

h = D/2 - (D/2)cos(omega*t)