1.A milling machine has a gear that spins, making one complete rotation counter clockwise every 360 milliseconds. The gear has a tooth with a marker in it as shown in the diagram below. There is a sensor 30mm below the center of the gear as viewed from the diagram that measures the height of the marker. The marker is supposed to start the rotation in the position shown on the left gear. Unfortunately the gears are out of sync and now the rotation starts in the position seen in the image at the right below.
The machinist has a table that tells him the height of the marker, but now needs a mathematician to help him. The gear is 15mm in diameter.
(a)Create two equations for the gear. The first models the height of the point in the gear as it was before the gears went out of sync.
(b)The second models the height of the marker in the machine when the gears are out of synchronization.
Create well-labelled graphs for each of these two situations that model the height of the marker on the gear.
(c)The machinist needs to know the height of the marker in the new and old gear positions at the following angles of rotation. Fill in the following table:
Angle of Rotation Old Marker Height New Marker Height
30°
60°
90°
120°
(d)The machinist needs one final piece of information. For maintenance, he needed to rotate the gear 240° to place the marker at a specific height. What height is this, and what rotation(s) on the new gear would provide the same height?
can anyone solve it
(a) To create the first equation for the gear, we need to determine the height of the marker as a function of time. We know that the gear makes one complete rotation clockwise every 360 milliseconds, which is equal to 2π radians.
Let's assume that at time t = 0, the marker is at its highest point (maximum height). As time progresses, the marker will move downwards along the gear.
We can model the height of the marker on the gear as a function of time using the equation:
h(t) = R - r * sin(ωt)
Where:
h(t) is the height of the marker at time t
R is the radius of the gear (half of the diameter, which is 7.5mm)
r is the radius of the marker (half of the diameter of the marker, which is 15mm)
ω is the angular velocity of the gear, given by ω = 2π/T, where T is the period of rotation. In this case, T = 360 milliseconds.
So, the equation becomes:
h(t) = 7.5 - 15 * sin((2π/0.36)t)
(b) To create the second equation for the gear when the gears are out of synchronization, we need to consider the new starting position of the marker. In this case, the marker is 30mm below the initial position shown in the diagram.
We can modify the equation to account for this new starting position:
h'(t) = 7.5 - 15 * sin((2π/0.36)t + φ)
Where φ is the phase shift, given by φ = sin^(-1)(30/15) = sin^(-1)(2) ≈ 1.5708 radians (or 90 degrees).
So, the equation becomes:
h'(t) = 7.5 - 15 * sin((2π/0.36)t + 1.5708)
(c) To find the marker height at specific angles of rotation, we can substitute the angles into our equations and solve for the height values.
For the old marker height (using equation h(t)):
Angle of Rotation Old Marker Height
30°
h(30) = 7.5 - 15 * sin((2π/0.36) * 30)
60°
h(60) = 7.5 - 15 * sin((2π/0.36) * 60)
90°
h(90) = 7.5 - 15 * sin((2π/0.36) * 90)
120°
h(120) = 7.5 - 15 * sin((2π/0.36) * 120)
For the new marker height (using equation h'(t)):
Angle of Rotation New Marker Height
30°
h'(30) = 7.5 - 15 * sin((2π/0.36) * 30 + 1.5708)
60°
h'(60) = 7.5 - 15 * sin((2π/0.36) * 60 + 1.5708)
90°
h'(90) = 7.5 - 15 * sin((2π/0.36) * 90 + 1.5708)
120°
h'(120) = 7.5 - 15 * sin((2π/0.36) * 120 + 1.5708)
(d) To find the height of the marker after rotating the gear 240°, we can substitute this angle into the equation h'(t) and solve for the height:
Height after rotating gear 240°:
h'(240) = 7.5 - 15 * sin((2π/0.36) * 240 + 1.5708)
To find the rotation(s) on the new gear that would provide the same height, we need to solve the equation h'(t) for the desired height:
Desired height = 7.5 - 15 * sin((2π/0.36)t + 1.5708)
Solving this equation will give you the rotation(s) on the new gear that correspond to the desired height.