The position of a car piston within the cylinder is described by the equation:
y = S sin(360
r t - 90
Where S is the stroke length of the cylinder, r is the rpm of the crank shaft, and t is
the time in minutes. If the stroke length is 2 inches and the crank shaft is running at
1300 rpm, find the position of the piston at t = 53 seconds. (remember that rpm
means revolutions per minute)
what's the problem?
y(t) = S sin(360rt-90)
S = 2
r = 1300
t = 53/60
y(53/60) = 2sin(360*1300*53/60-90)
= 2sin(1148.3333*360 - 90)
now, we can toss out all multiples of 360, so we are left with
y = 2sin(240-90) = 2sin(150) = 2(.5) = 1.0
oops.
y = 2sin(120-90) = 2sin(30) = 2(.5) = 1.0
To find the position of the piston at t = 53 seconds, we need to substitute the given values of S, r, and t into the equation y = S sin(360r t - 90) and solve for y.
Given:
S = 2 inches (stroke length)
r = 1300 rpm (revolutions per minute)
t = 53 seconds
First, we need to convert the given time t from seconds to minutes:
53 seconds = 53/60 minutes ≈ 0.8833 minutes
Now we can substitute the values into the equation:
y = 2 sin(360 * 1300 * 0.8833 - 90)
Let's calculate the expression inside the sine function:
360 * 1300 * 0.8833 - 90 ≈ 410,171.94
Substituting this value back into the equation:
y = 2 sin(410,171.94)
Since the sine function takes the angle in radians, we need to convert 410,171.94 degrees to radians:
410,171.94° = (410,171.94 * π) / 180 radians ≈ 7153.41 radians
So the equation becomes:
y = 2 sin(7153.41)
Now, calculate the sine value:
y = 2 * sin(7153.41) ≈ 1.697 inches
Therefore, at t = 53 seconds, the position of the car piston within the cylinder is approximately 1.697 inches.