Question

A 300 -newton weight rests on a smooth (friction negligi- ble) inclined plane that makes an angle of \(30^{\circ}\) with the horizon- tal. 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.

Short answer

The force needed is 150 N parallel to the plane.

Step-by-step solution

Understand the Problem

We need to find the force parallel to the inclined plane that will prevent a 300-N weight from sliding down. The plane is at a 30-degree angle to the horizontal, and we assume there is no friction.

Analyze the Forces

The gravitational force can be split into two components: one perpendicular to the plane and one parallel to the plane. The parallel component is what causes the object to slide down.

Calculate the Parallel Force Component

The force parallel to the plane is given by the formula: \[ F_{parallel} = mg \sin(\theta) \] where \( m = 300 \text{ N} \) is the weight and \( \theta = 30^{\circ} \).

Substitute Values and Solve

Substitute the known values into the equation: \[ F_{parallel} = 300 \times \sin(30^{\circ}) \] We know \( \sin(30^{\circ}) = 0.5 \). So, \[ F_{parallel} = 300 \times 0.5 = 150 \text{ N} \]

Conclusion

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

Key concepts

Inclined Plane

An inclined plane is a flat surface that is tilted at an angle, not horizontal or vertical. In physics, it is a simple machine that helps to move objects across vertical heights with less effort. By inclining a surface, the amount of force required to lift an object is reduced. The object moves along the slope instead of against the gravitational force acting directly downwards.
Inclined planes are commonly encountered in daily life. Examples include ramps, stairs, and hills. Understanding how forces act on objects on these surfaces is important, especially when determining how much force is needed to move or keep an object stationary. With inclined planes, the gravitational force acting on an object can be split into different components, which is where our next topic comes in.

Force Components

When an object is on an inclined plane, the force of gravity acting on it can be divided into two distinct components:

• Parallel to the plane
• Perpendicular to the plane

The component parallel to the plane (\( F_{parallel} \) ) causes the object to slide down the slope. In contrast, the perpendicular component is what presses the object against the surface. In physics, resolving forces into components is a common technique. It allows us to better analyze and understand the impact of forces acting on objects. By resolving the gravitational force into these components, you can calculate how much of the force affects the object's motion down or against the plane.

Gravitational Force

Gravitational force is the force that the Earth exerts on an object. It acts downward towards the center of the Earth and is proportional to the object's mass. The equation used to calculate this force is:\[ F = mg \]where \(m\) is the mass of the object and \(g\) is the acceleration due to gravity, approximately \(9.8 \text{ m/s}^2\).
On inclined planes, gravity influences the movement of an object. Importantly, it must be decomposed into two components as discussed earlier. The parallel component of gravitational force is calculated by considering the angle of the incline and is critical in determining if an object will slide down or stay put. By understanding gravitational force, we can predict how objects will behave on slopes.

Trigonometry in Physics

In the context of physics, trigonometry is crucial for resolving forces into components. It involves using sine, cosine, and tangent functions to calculate the sizes of force components. In the case of an inclined plane, we use the sine function to find the parallel force:\[ F_{parallel} = mg \sin(\theta) \]Here, \(\theta\) is the angle of the incline. The sine of this angle tells us the fraction of the gravitational force acting parallel to the slope. For example, when \(\theta = 30^{\circ}\), the sine is \(0.5\). Thus, half of the gravitational force pulls the object down the ramp.
Understanding trigonometry enhances problem-solving skills in physics. It allows you to break down complex force scenarios into manageable calculations, leading to clearer insights into how objects will move or stay static on inclined planes.