To solve this problem, we will need to use the equations of motion for a pendulum. The motion of a pendulum can be described by the following equation:
a = (-g/L) * sin(theta)
where:
a = acceleration of the bob
g = acceleration due to gravity (32.2 ft/s^2)
L = length of the cable supporting the bob
theta = angle between the cable and the vertical
To find the total acceleration of the bob, we need to find the horizontal and vertical components of the acceleration. We can use the given velocity to find the horizontal component of the acceleration:
a_h = v^2 / L
a_h = (15 ft/s)^2 / 5 ft
a_h = 45 ft/s^2
To find the vertical component of the acceleration, we need to find the angle between the cable and the vertical. We can use trigonometry to find this angle:
sin(theta) = opposite / hypotenuse
sin(theta) = 5 ft / 10 ft
theta = sin^-1(0.5)
theta = 30 degrees
Now we can use the equation of motion for a pendulum to find the total acceleration of the bob:
a = (-g/L) * sin(theta)
a = (-32.2 ft/s^2 / 5 ft) * sin(30 degrees)
a = -16.1 ft/s^2
The total acceleration of the bob is the vector sum of the horizontal and vertical components:
a_total = sqrt(a_h^2 + a_v^2)
a_total = sqrt((45 ft/s^2)^2 + (-16.1 ft/s^2)^2)
a_total = 48.3 ft/s^2
To find the tension in the cable, we can use Newton's second law:
T - mg = ma
where:
T = tension in the cable
m = mass of the bob (10 lb)
g = acceleration due to gravity (32.2 ft/s^2)
a = total acceleration of the bob
Substituting the values we have found:
T - (10 lb)(32.2 ft/s^2) = (10 lb)(48.3 ft/s^2)
T = 226 lb
Therefore, the tension in the cable is 22.6 lb.