Question

additional topics in trigonometry
finding magnitudes of forces related to an object on a ramp

a package is on a ramp that is inclined \\(36^\circ\\) to the horizontal, as shown.
to prevent the package from sliding, a force \\(\mathbf{q}\\) of magnitude \\(105\text{ 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 \\(\mathbf{c}\\) of the package against the ramp.

(b) find the weight of the package, represented by force \\(\mathbf{g}\\).

Explanation:

Identify the physical system and forces

Using the Incline Plane Forces and Static Equilibrium knowledge points

• Ramp angle: \(\theta = 36^\circ\)
• Force parallel to the ramp preventing sliding: \(Q = 105\text{ N}\) (directed up the ramp)
• Gravity force: \(\mathbf{G}\) (directed vertically downward, magnitude \(W\))
• Force of the package against the ramp: \(\mathbf{C}\) (directed perpendicular to the ramp, magnitude \(C\))
• Force pulling down the ramp: \(\mathbf{P}\) (magnitude \(P\))

Relate forces using static equilibrium

Using the Static Equilibrium knowledge point

$$
LATEXBLOCK0
$$

Since the package is prevented from sliding by force \(\mathbf{Q}\), we have:

$$ Q = P = 105\text{ N} $$

Calculate the weight of the package (force G)

Using the Incline Plane Forces knowledge point

$$
LATEXBLOCK1
$$

Calculate the force of the package against the ramp (force C)

Using the Incline Plane Forces knowledge point

$$
LATEXBLOCK2
$$

Answer:

Question a

(a) Find the magnitude of the force \(\mathbf{C}\) of the package against the ramp.

\(144.52\text{ N}\)

Question b

(b) Find the weight of the package, represented by force \(\mathbf{G}\).

\(178.64\text{ N}\)