Question

a package is on a ramp that is inclined 32° to the horizontal, as shown. to prevent the package from sliding, a force q of magnitude 117 n is required. the figure is not drawn to scale. complete the following. do not round any intermediate computations, and round your answers to the nearest hundredth. (a) find the magnitude of the force c of the package against the ramp. (b) find the weight of the package, represented by force g.

Explanation:

Step1: Analyze the forces

When a package is on an inclined - plane, the force \(Q\) (the force required to prevent sliding) is related to the weight \(G\) of the package by \(Q = G\sin\theta\), and the force \(C\) (the force against the ramp) is related to the weight \(G\) by \(C = G\cos\theta\), where \(\theta = 32^{\circ}\).

Step2: Find the weight \(G\)

Given \(Q = 117\space N\) and \(Q = G\sin\theta\) (\(\theta = 32^{\circ}\)), we can solve for \(G\).

$$G=\frac{Q}{\sin\theta}=\frac{117}{\sin(32^{\circ})}$$

Since \(\sin(32^{\circ})\approx0.5299\), then \(G=\frac{117}{0.5299}\approx220.80\space N\)

Step3: Find the force \(C\)

Since \(C = G\cos\theta\) and \(G=\frac{117}{\sin(32^{\circ})}\), then \(C=\frac{117\cos(32^{\circ})}{\sin(32^{\circ})}\)
We know that \(\cos(32^{\circ})\approx0.8480\) and \(\sin(32^{\circ})\approx0.5299\)

$$C = 117\times\frac{0.8480}{0.5299}=117\times1.60\approx187.20\space N$$

Answer:

(a) \(187.20\space N\)
(b) \(220.80\space N\)