Concept explainers
Belts and Pulleys A thin belt of length L surrounds two pulleys of radii R and r, as shown in the figure to the right.
- (a) Show that the angle θ (in rad) where the belt crosses itself satisfies the equation
[Hint: Express L in terms of R, r, and θ by adding up the lengths of the curved and straight parts of the belt.]
- (b) Suppose that R = 2.42 ft, r = 1.21 ft, and L = 27.78 ft. Find θ by solving the equation in part (a) graphically. Express your answer both in radians and in degrees.
(a)
To show: The equation
Explanation of Solution
Given:
The thin belt of length L surrounds two pulleys of radii R and r, as shown in the given figure.
Calculation:
Redraw the given figure for convenient as shown below in Figure 1.
Obtain the angle of
Now, find the angle of
Note that, the arc length is
Obtain the arc length of AB as shown below.
Similarly, obtain the arc length of CD as follows.
Now, obtain the length of the bell on left side as shown below.
Similarly, obtain the length on the right side as follows.
Now consider the triangle AFE,
Similarly,
Note that, the length of the belt is denoted by L and it can be calculated as follows.
On further simplification,
Hence, it is proved.
b.
The angle
Answer to Problem 66E
The values of
Explanation of Solution
From part (a),note that
Substitute the given values
Use online graph calculator and obtain the graph of the function
From Figure, it is observed that the values of
Therefore, the values of
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