.Pump Jack Problem: An oil well pump jack is

shown in Figure 3-9f. As the motor turns, the
walking beam rocks back and forth, pulling the
rod out of the well and letting it go back into
the well. The connection between the rod and
the walking beam is a steel cable that wraps
around the cathead. The distanced from the
ground to pointP, where the cable connects to
the rod, varies periodically with time.
a.As the walking beam rocks, the angleθ it
makes with the ground varies sinusoidally
with time. The angle goes from a minimum
ofD0.2 radian to a maximum of 0.2 radian.
How many degrees correspond to this range
of angle (θ)?
b.The radius of the circular arc on the cathead
is 8 ft. What arc length on the cathead
corresponds to the range of angles in part a?
c.The distance,d, between the cable-to-rod
connector and the ground varies
sinusoidally with time. What is the
amplitude of the sinusoid?
d. Suppose that the pump is started at time t= 0 s. One second later, Pis at its highest point above the ground.P is at its next low
point 2.5 safter that. When the walking
beam is horizontal, pointP is 7 ft above the
ground. Sketch the graph of this sinusoid.
e.Find a particular equation expressingd as a function oft.
f.How far above the ground isP att = 9?

Suppose that the pump is started at time t= 0 s. One second later, Pis at its highest point above the ground.P is at its next low

point 2.5 safter that. When the walking
beam is horizontal, pointP is 7 ft above the
ground. Sketch the graph of this sinusoid.

a. To find the number of degrees corresponding to the range of angle θ, we can use the fact that 1 radian is equal to 180 degrees. So, the range of angles in radians can be converted to degrees by multiplying by 180/π. In this case, the range is 0.4 radians (0.2 + 0.2), so the equivalent range in degrees is (0.4 * 180/π) degrees.

b. To find the arc length on the cathead corresponding to the range of angles, we can use the formula for finding the length of a circular arc. The formula is: arc length = radius * angle (in radians). In this case, the radius is given as 8 ft and the range of angles is 0.4 radians. So, the arc length on the cathead is 8 ft * 0.4 radians.

c. To find the amplitude of the sinusoidal variation in distance d, we can use the fact that the amplitude of a sinusoid is half the range of variation. In this case, the range of angles is given as 0.2 radians, so the amplitude of the sinusoid is 0.2/2 radians.

d. Based on the given information, we can sketch the graph of this sinusoid. First, we know that point P is at its highest point above the ground when t = 0 s. Since the walking beam is horizontal at this point, P is 7 ft above the ground. We also know that P is at its next low point 2.5 s after that. From this information, we can conclude that the period of the sinusoid is 2.5 s. We can now sketch the graph of the sinusoid using the amplitude and period.

e. To find a particular equation expressing d as a function of t, we need more information about the specific form of the sinusoidal variation. Without that information, we cannot provide a particular equation.

f. To find how far above the ground P is at t = 9, we need the equation expressing d as a function of t. Without that equation, we cannot determine the value of d at t = 9.