Question

question 7
a 30-pound box is resting on a ramp that is inclined 7°. rounding to 2 decimal places:

a. find the magnitude of the component of the force that is normal (perpendicular) to the ramp.
pounds

b. find the magnitude of the component of the force that is parallel to the ramp.
pounds

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Explanation:

Step1: Recall Force Components on Incline

When a weight \( F = 30 \) pounds rests on an inclined ramp at angle \( \theta = 7^\circ \), the normal (perpendicular) component \( F_n \) and parallel component \( F_p \) of the weight (force due to gravity) are given by:

• Normal component: \( F_n = F \cos\theta \) (adjacent side in right triangle, since normal is perpendicular to ramp, angle between weight vector and normal is \( \theta \))
• Parallel component: \( F_p = F \sin\theta \) (opposite side, parallel to ramp)

Step2: Calculate Normal Component (Part a)

Substitute \( F = 30 \), \( \theta = 7^\circ \) into \( F_n = 30 \cos(7^\circ) \).
First, convert calculator to degree mode. Compute \( \cos(7^\circ) \approx 0.992546 \).
Then, \( F_n = 30 \times 0.992546 \approx 29.77638 \). Round to 2 decimal places: \( 29.78 \) pounds.

Step3: Calculate Parallel Component (Part b)

Substitute \( F = 30 \), \( \theta = 7^\circ \) into \( F_p = 30 \sin(7^\circ) \).
Compute \( \sin(7^\circ) \approx 0.121869 \).
Then, \( F_p = 30 \times 0.121869 \approx 3.65607 \). Round to 2 decimal places: \( 3.66 \) pounds.

Answer:

a. \( 29.78 \)
b. \( 3.66 \)