Problem 19 - Calculus Early Transcendentals: Pearson New International Edition

Question

A 300 -newton weight rests on a smooth (friction negligible) inclined plane that makes an angle of \(30^{\circ}\) with the horizontal. What force parallel to the plane will just keep the weight from sliding down the plane? Hint: Consider the downward force of 300 newtons to be the sum of two forces, one parallel to the plane and one perpendicular to it.

Step-by-Step Solution

Answer

150 N parallel to the plane prevents sliding.

Step 1: Understand the Problem

To find the force that keeps the weight from sliding, we need to resolve the weight into two components: one parallel and one perpendicular to the inclined plane.

Step 2: Identify the Forces Involved

The weight of 300 N acts vertically downward. The component of this force perpendicular to the plane will not affect sliding. We're interested in the component parallel to the inclined plane since this component drives the sliding movement.

Step 3: Determine the Parallel Component of Force

The parallel component of force can be calculated using the formula: \( F_{\parallel} = W \cdot \sin(\theta) \), where \( W \) is the weight and \( \theta \) is the angle of the incline.

Step 4: Calculate the Parallel Force

Use the values given: \( W = 300 \text{ N} \) and \( \theta = 30^{\circ} \). Thus, \( F_{\parallel} = 300 \cdot \sin(30^{\circ}) = 300 \cdot 0.5 = 150 \text{ N} \).

Step 5: Conclusion

The force parallel to the plane required to keep the weight from sliding is 150 N.

Key Concepts

Inclined Plane

An inclined plane is a flat surface tilted at an angle to the horizontal. It is a simple machine that helps in moving loads with less effort. When an object is placed on the inclined plane, gravity pulls it downward along the plane. If the surface is smooth, most of the force will act parallel to the surface, causing potential sliding.
Inclined planes are often seen in everyday environments, such as wheelchair ramps and slides. By understanding the forces at play, one can predict how and why objects slide or remain stationary. Key forces include gravity and normal force. Grasping these concepts is essential to solve many physics problems, particularly those related to motion and equilibrium on slopes.

Force Components

Force components are crucial for analyzing forces acting on objects, especially when dealing with inclined planes. When a weight rests on an inclined surface, it experiences forces that can be split into two components. These components help us understand how forces behave in different directions.

• Vertical Component: This force component acts perpendicular to the plane. It is unaffected by the incline's slope and contributes to the normal force.
• Parallel Component: This component acts along the plane and is responsible for any possible sliding of the object down the slope.

Determining these components requires breaking down the original force vector (the weight of the object) using trigonometry. This breakdown is essential for solving problems where motion or balance on an incline is involved.

Trigonometry in Physics

Trigonometry is an indispensable tool in physics, especially for resolving forces. In the context of inclined planes, trigonometry allows us to decompose forces into components that align with the axes of interest: parallel and perpendicular to the inclined surface.
The sine and cosine functions are particularly useful.

• Sine function: Indicates the ratio of the length of the opposite side to the hypotenuse, which helps in calculating the parallel component of force when dealing with inclined planes.
• Cosine function: Represents the adjacent side's length to the hypotenuse, used to find components perpendicular to the inclined surface.

For the given problem, the angle of inclination is 30°, and using sine, we find the parallel force component: \[F_{\parallel} = F \cdot \sin(\theta)\]where \( F \) is the force due to weight and \( \theta \) is the angle. This method allows precise calculations of forces in physics, making problem-solving much more manageable.