An air-conditioning unit is fastened to a roof that slopes at an angle of 35° above the horizontal (Fig. 1.34). Its weight is a force on the air conditioner that is directed vertically downward. In order that the unit not crush the roof tiles, the component of the unit’s weight perpendicular to the roof cannot exceed 425 N. (One newton, or 1 N, is the SI unit of force. It is equal to 0.2248 lb.) (a) What is the maximum allowed weight of the unit? (b) If the fasteners fail, the unit slides 1.50 m along the roof before it comes to a halt against a ledge. How much work does the weight force do on the unit during its slide if the unit has the weight calculated in part (a)? The work done by a force on an object that undergoes a displacement is
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SOLUTION GUIDE
IDENTIFY and SET UP
1. This problem involves vectors and components. What are the known quantities? Which aspect(s) of the weight vector (magnitude, direction, and/or particular components) represent the target variable for part (a)? Which aspect(s) must you know to solve part (b)?
2. Make a sketch based on Fig. 1.34. Draw the x- and y- axes, choosing the positive direction for each. Your axes don’t have to be horizontal and vertical, but they do have to be mutually perpendicular. Figure 1.34 shows a convenient choice of axes: The x -axis is parallel to the slope of the roof.
3. Choose the equations you’ll use to determine the target variables.
EXECUTE
4. Use the relationship between the magnitude and direction of a vector and its components to solve for the target variable in
part (a). Be careful: Is 35° the correct angle to use in the equation? (Hint: Check your sketch.)
5. Make sure your answer has the correct number of significant figures.
6. Use the definition of the scalar product to solve for the target variable in part (b). Again, use the correct number of significant figures.
University Physics 14th Edition Download
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EVALUATE
7. Did your answer to part (a) include a vector component whose absolute value is greater than the magnitude of the vector? Is that possible?
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University Physics 14th Edition Answer Key
8. There are two ways to find the scalar product of two vectors, one of which you used to solve part (b). Check your answer by repeating the calculation, using the other way. Do you get the same answer?