A man is dragging a trunk up the loading ramp of a mover's truck. The ramp has a slope angle of $20.0$°, and the man pulls upward with a force $\overrightarrow{F}$ whose direction makes an angle of $30.0$° with the ramp (Fig. E$4.4$). How large will the component $F_y$ perpendicular to the ramp be then?
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Verified step by step guidance

1. Step 1: Identify the given angles and forces. The ramp has a slope angle of 20.0°, and the force F→ makes an angle of 30.0° with the ramp. The goal is to find the perpendicular component Fy of the force F→ relative to the ramp.
2. Step 2: Understand the relationship between the force components. The force F→ can be broken into two components: one parallel to the ramp (Fx) and one perpendicular to the ramp (Fy). Fy is given by Fy = F * sin(θ), where θ is the angle between the force and the ramp.
3. Step 3: Substitute the angle θ into the formula. Here, θ = 30.0° because the force F→ makes an angle of 30.0° with the ramp. Thus, Fy = F * sin(30.0°).
4. Step 4: Recall the trigonometric value for sin(30.0°). The sine of 30.0° is 0.5. Therefore, Fy = F * 0.5.
5. Step 5: To find Fy, multiply the magnitude of the force F by 0.5. This will give the perpendicular component of the force relative to the ramp.

Key Concepts

Components of Force

Forces can be broken down into components that act along the axes of a coordinate system. In this scenario, the pulling force F can be resolved into two components: one parallel to the ramp and one perpendicular to it. Understanding how to decompose forces into their components is essential for analyzing the effects of the applied force on the trunk's motion.

Trigonometric Functions

Trigonometric functions, such as sine, cosine, and tangent, are crucial for relating angles to the ratios of the sides of right triangles. In this problem, the angles given (30° and 20°) can be used with these functions to calculate the components of the force acting on the trunk. For example, the perpendicular component can be found using the sine function.

Newton's Laws of Motion

Newton's Laws of Motion describe the relationship between the motion of an object and the forces acting on it. The first law states that an object at rest will remain at rest unless acted upon by a net force. In this context, understanding how the applied force and the gravitational force interact on the ramp is essential for determining the trunk's acceleration and the forces involved in moving it.