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Chapter 12: Problem 40

Three forces act on an object. Two of the forces are at an angle of \( 100^\circ \) to each other and have magnitudes 25 N and 12 N. The third is perpendicular to the plane of these two forces and has magnitude 4 N. Calculate the magnitude of the force that would exactly counterbalance these three forces.

Short Answer

Expert verified
The magnitude of the counterbalancing force is 26.29 N.

Step by step solution

01

Identify Known Values

We have three forces: 1. Force A with a magnitude of 25 N. 2. Force B with a magnitude of 12 N, at an angle of \(100^\circ\) to Force A. 3. Force C is perpendicular to the plane containing Forces A and B with a magnitude of 4 N. These three forces form a 3D system.
02

Resolve Forces A and B

Since Forces A and B are in a plane, we break them down into x and y components:- The x-component of Force A is \( 25 \cos(0^\circ) = 25 \) N.- The y-component of Force A is \( 25 \sin(0^\circ) = 0 \) N.- The x-component of Force B is \( 12 \cos(100^\circ) \approx -2.09 \) N.- The y-component of Force B is \( 12 \sin(100^\circ) \approx 11.82 \) N.
03

Calculate Resultant Force in Plane

Add the x and y components of Forces A and B:- Total x-component: \( 25 - 2.09 = 22.91 \) N.- Total y-component: \( 0 + 11.82 = 11.82 \) N.The resultant force in the plane is calculated using the Pythagorean theorem:\[ \sqrt{(22.91)^2 + (11.82)^2} \approx 25.98 \text{ N} \]
04

Combine with Perpendicular Force

Since Force C is perpendicular to the resultant of Forces A and B, we can use the Pythagorean theorem to find the combined force:\[ \sqrt{(25.98)^2 + (4)^2} \approx 26.29 \text{ N} \]
05

Determine Counterbalance Force

The counteracting force must equal in magnitude and opposite in direction to the resultant force. Therefore, the counterbalance force needed is 26.29 N.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

3D Vector Components
When dealing with forces acting on an object, especially in a three-dimensional space, it becomes essential to break down each force into its components. These components allow us to visualize and calculate the effects of each force more accurately.
For forces in a 3D system, we often use Cartesian coordinates, which consist of the x, y, and z axes. Each force can have components along these three axes, depending upon its direction.
  • Force A, with a magnitude of 25 N, was aligned along the x-axis in this example, making its y-component zero.
  • Force B, with a magnitude of 12 N, was at an angle and therefore had non-zero components both along the x and y axes, calculated using trigonometric functions like cosine and sine.
  • Force C, with a magnitude of 4 N, was acting perpendicular to the plane formed by Forces A and B, hence occupying a position along the z-axis.
Using these 3D vector components, one can analyze the interactions between the forces in much more detail, allowing for an accurate determination of the total or resultant force acting on the system.
Force Balance
Force balance is a fundamental concept in physics, especially in mechanics. It refers to the idea of finding a state in which all forces acting on an object are counterbalanced, resulting in zero net force.
  • In the context of the exercise, three forces were acting on an object. The goal was to find another force that could exactly counterbalance these three.
  • Force A and Force B, both lying in the same plane, were summed up to resolve their combined effect in the x and y directions.
  • Force C, being perpendicular, influenced the resultant force along the z-axis.
By calculating the resultant forces from these components, we could find the necessary conditions or an additional force needed to bring the net force on the object to zero, achieving perfect force balance.
Resultant Force
The concept of a resultant force is vital for understanding how multiple forces interact and affect an object. The resultant force is the single force that has the same effect on an object as all the original forces combined.
  • For this problem, we first calculated the resultant force in the plane from Forces A and B utilizing components and the Pythagorean theorem.
  • After finding this planar resultant, we then considered Force C which was orthogonal to this plane, adding another dimension to the resultant calculation.
  • The final overall resultant force was again calculated using the Pythagorean theorem, now incorporating Force C's influence.
Knowing how to compute the resultant force is crucial when assessing situations involving multiple forces, as it provides a clear picture of the overall force affecting an object, allowing for better planning and adjustment of necessary counteracting forces.

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