Х MIRANDA ALVARAD… Dynamics | 2 Forces On Inclined Planes NAME Scenario Angela And Carlos Are Asked To Determine The Relationship Between The Normal Force On A Box Of Mass And The Angle Of Incline Of The Box As The Box Sits Of Rest On The Incline. Using Representations MARTA: The Dot At Night Represents TheBlock On The Indline. Dawa Free Body Diagram
The correct answer and explanation is:
In this scenario, Angela and Carlos are asked to determine the relationship between the normal force on a box and the angle of incline. The box sits at rest on an inclined plane, and the objective is to understand how the angle of inclination affects the normal force acting on the box.
Key Concepts:
- Normal Force (N): This is the force exerted by a surface to support the weight of an object resting on it. It acts perpendicular to the surface. When the box is on an inclined plane, the normal force is not equal to the weight of the box but is instead influenced by the angle of the incline.
- Weight (W): The weight of the box is the force due to gravity acting on its mass and is directed vertically downward. The magnitude of weight is given by W=mgW = mg, where mm is the mass of the box and gg is the acceleration due to gravity.
- Decomposition of Forces: The weight of the box can be decomposed into two components:
- One component parallel to the incline (W∥=mgsinθW_{\parallel} = mg \sin \theta) that causes the box to potentially slide down the incline.
- Another component perpendicular to the incline (W⊥=mgcosθW_{\perp} = mg \cos \theta) that is counteracted by the normal force.
Relationship Between Normal Force and Angle of Incline:
The normal force (NN) is equal to the perpendicular component of the weight of the box, which is N=mgcosθN = mg \cos \theta, where θ\theta is the angle of incline. As the angle of incline increases, the cosine of the angle decreases, meaning that the normal force decreases as the incline becomes steeper.
Explanation:
As the angle of the incline increases, the weight of the box is distributed differently between the components parallel and perpendicular to the surface. The normal force, which opposes the component of the weight acting perpendicular to the surface, decreases as the angle increases. This is because more of the weight is acting along the direction of the incline, and less is acting perpendicular to it.
In conclusion, the normal force on the box decreases as the angle of incline increases. This relationship is directly influenced by the cosine of the incline angle. When the incline angle is 0° (the plane is horizontal), the normal force equals the weight of the box, and as the angle increases toward 90°, the normal force approaches zero.