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Chapter 5: Problem 21

Two people, one at each end of a uniform beam that weighs \(400 \mathrm{~N}\), hold the beam at an angle of \(25.0^{\circ}\) to the horizontal. How large a vertical force must each person exert on the beam?

Short Answer

Expert verified
Each person exerts a vertical force of about 220.7 N.

Step by step solution

01

Understand the Problem

We have a uniform beam weighing 400 N, held at an angle of 25掳 to the horizontal by two people at each end. We need to find out the vertical force each person must exert to hold the beam in equilibrium.
02

Analyze Forces

The weight of the beam acts at its center of gravity, which is the midpoint of the beam. Therefore, the weight, 400 N, is distributed equally to both sides of the beam, with each person ultimately supporting half the weight of the beam, which is 200 N, at the midpoint vertically.
03

Calculate the Effective Vertical Force

Since the beam is held at an angle of 25掳 to the horizontal, not all of the weight acts vertically. To find the vertical component of the force each person exerts, we use trigonometry: each person needs to exert a force equal to half the weight of the beam divided by the cosine of the angle.The vertical component of the force each person exerts is given by:\[F = \frac{400 \text{ N}}{2} / \cos(25^{\circ})\]
04

Compute the Result

Plug the values in the equation:\[F = \frac{200 \text{ N}}{\cos(25^{\circ})}\]Calculate \(\cos(25^{\circ})\):\[\cos(25^{\circ}) \approx 0.9063\]Thus:\[F = \frac{200}{0.9063} \approx 220.7 \text{ N}\]
05

Conclusion

Each person must exert a vertical force of approximately 220.7 N to hold the beam in equilibrium.

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Key Concepts

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

Vertical Force Calculation
When dealing with equilibrium in physics, understanding how to calculate vertical force is crucial. In this exercise, we have a beam that weighs 400 Newtons. The beam is held by two people at each end, and you need to determine the specific vertical force each person must exert. Here's how we go about it.

First, recognize that the beam's weight exerts a force due to gravity, which acts directly downward through the beam's center of gravity. Since the beam is uniform, its center of gravity is in the middle. Therefore, the 400 N force is equally split between the two people holding the beam. This means each individual supports 200 N.

However, the beam is angled, which complicates the force calculation. To find the true vertical force, we have to adjust this 200 N force to account for the angle involved. That's where trigonometry comes into play, allowing us to calculate the vertical component of this force effectively.
Trigonometry in Physics
Trigonometry is a powerful tool used in physics to resolve forces acting at angles. In this scenario, the beam is held at a 25-degree angle to the horizontal, meaning not all of the weight acts vertically. We use trigonometric functions to find the vertical component of the force.

To determine the vertical force each person must apply, we need to use the cosine function. The cosine of an angle in right-angle trigonometry relates the adjacent side of a triangle (which is, in this case, the total vertical force required) to the hypotenuse (the actual force exerted).

Mathematically, this is expressed as:
  • Force required per person: 200 N
  • Angle with the horizontal: 25掳
Using the formula:
\[ F = \frac{200 \text{ N}}{\cos(25^{\circ})} \]
The result gives us the vertical force component needed from each person. This approach ensures they are applying the correct force to maintain equilibrium.
Force Components
Understanding force components is vital when analyzing any physical system where forces are at angles. In any situation where a beam or object is held at an angle, the original force (like the 400 N weight in our example) can be broken down into components: a vertical component and a horizontal component.

Each of these components affects the beam differently, dependent on their direction. The vertical component is of primary interest here as it is what counters the gravitational force to keep the beam in equilibrium.

The breakdown of forces into components is done using basic trigonometric principles:
  • Vertical component: \( F_y \), calculated using the cosine of the angle
  • Horizontal component: \( F_x \), usually calculated using the sine of the angle
Only the vertical component directly combats gravity to support the beam, illustrating why we focus on this calculation when seeking equilibrium conditions. By resolving the force into its components, we can correctly determine how much force needs to be exerted in the desired direction.

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Most popular questions from this chapter

An essentially weightless shovel is \(120 \mathrm{~cm}\) long. Someone holds it horizontally, supporting it with his left hand at the shovel's center of gravity and his right hand \(80.0 \mathrm{~cm}\) to the right of the \(\mathrm{c.g}\). The shovel contains a 20.0-N rock whose c.g. is \(8.00 \mathrm{~cm}\) to the right of the edge of the shovel. How much force does the person exert down on the handle? [Hint: Draw a diagram and take the torques around the left hand to avoid the force of the left hand.]

A horizontal essentially weightless lever is pivoted so it can rotate freely in a vertical plane. A downward force of \(30.0 \mathrm{~N}\) is applied perpendicularly to the lever at a point \(25.0 \mathrm{~cm}\) from and to the right of the pivot. Determine the torque on the lever, about the pivot. [Hint: Draw a diagram and specify the direction of the torque.]

A horizontal essentially weightless lever is pivoted at its center so it can rotate freely in a vertical plane. A downward force of \(80.0 \mathrm{~N}\) is applied perpendicularly to the lever at a point \(35.0 \mathrm{~cm}\) from and to the right of the pivot. Another downward force of \(100.0 \mathrm{~N}\) is applied perpendicularly to the lever at a point \(15.0 \mathrm{~cm}\) from and to the left of the pivot. Determine the net torque on the lever. [Hint: Draw a diagram.]

A seesaw is \(5.00 \mathrm{~m}\) long with a fulcrum at its center. The uniform plank is balanced horizontally when a \(40.0\) -kg kid sits at the very end on the right and an \(80.0\) -kg kid sits somewhere on the left. Locate that second kid. [Hint: Draw a diagram.]

A force of \(1000 \mathrm{~N}\) is applied downward at the right end of a \(1.50-\mathrm{m}\) long, essentially weightless horizontal crowbar. The bar is pivoted on a rock \(1.25 \mathrm{~m}\) from the right end. What is the maximum amount of weight that can be supported on the left end before the bar moves? [Hint: Draw a diagram. Watch out for significant figures.]
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