Question

In a physics laboratory class, three massless ropes are tied together at a point. A pulling force is applied along each rope: \(F_{1}=150 . \mathrm{N}\) at \(60.0^{\circ}, F_{2}=200 . \mathrm{N}\) at \(100 .^{\circ}, F_{3}=100 . \mathrm{N}\) at \(190 .^{\circ} .\) What is the magnitude of a fourth force and the angle at which it acts to keep the point at the center of the system stationary? (All angles are measured from the positive \(x\) -axis.)

Short answer

Short Answer: To find the magnitude and direction of the fourth force needed to keep the system stationary, follow these steps: 1. Express each given force as a vector with components in x and y directions using trigonometry. 2. Find the total x and y components of the forces acting on the system by summing the individual components. 3. Calculate the x and y components of the fourth force required to make the total force in both x and y directions zero. 4. Find the magnitude and direction of the fourth force using the calculated components and the arctan function. 5. Convert the direction to the appropriate angle with respect to the positive x-axis, ensuring it's consistent with the calculated components of the fourth force.

Step-by-step solution

Express each force as a vector with components in x and y directions.

To find the components of \(\vec{F}_{1}\), we use basic trigonometry: - \(F_{1x} = F_{1} \cos{60^{\circ}} = 150 \cos{60^{\circ}}\) - \(F_{1y} = F_{1} \sin{60^{\circ}} = 150 \sin{60^{\circ}}\) For \(\vec{F}_{2}\): - \(F_{2x} = F_{2} \cos{100^{\circ}} = 200 \cos{100^{\circ}}\) - \(F_{2y} = F_{2} \sin{100^{\circ}} = 200 \sin{100^{\circ}}\) For \(\vec{F}_{3}\): - \(F_{3x} = F_{3} \cos{190^{\circ}} = 100 \cos{190^{\circ}}\) - \(F_{3y} = F_{3} \sin{190^{\circ}} = 100 \sin{190^{\circ}}\)

Find the total x and y components of the forces acting on the system.

Now that we have the components of each force, we will sum the components in the x and y directions. - \(F_{x(total)} = F_{1x} + F_{2x} + F_{3x}\) - \(F_{y(total)} = F_{1y} + F_{2y} + F_{3y}\)

Calculate the fourth force needed to keep the system stationary.

Since the system is to be kept stationary, the total force in x and y directions should be zero. - \(F_{4x} = -F_{x(total)}\) - \(F_{4y} = -F_{y(total)}\)

Calculate the magnitude and direction of the fourth force.

Now that we have the x and y components of the fourth force, the magnitude and the angle can be calculated by: - Magnitude: \(F_{4} = \sqrt{F_{4x}^2 + F_{4y}^2}\) - Direction: \(\theta_{4} = \arctan{\frac{F_{4y}}{F_{4x}}}\)

Convert the direction to the appropriate angle with respect to the positive x-axis.

Since the angle reported by the arctan function can be ambiguous, make sure to verify and correct \(\theta_{4}\) if needed, to ensure the angle is consistent with the direction of the calculated components of \(\vec{F}_{4}\). The magnitude and direction of the force \(\vec{F}_4\) will keep the system stationary.

Key concepts

Vectors

In physics, vectors are essential because they help us understand and describe quantities that have both direction and magnitude. A vector is usually represented by an arrow. The arrow's length represents the vector's magnitude, and the arrow's direction represents the vector direction.

When dealing with forces like in our exercise, it's crucial to express them as vectors. This means noting not just how strong the force (magnitude) is, but also in which direction it acts. This becomes particularly useful when combining forces, as vectors allow us to break the forces into components along different axes for better analysis. In our exercise, forces are broken down along the x and y axes for easier handling.

Trigonometry

Trigonometry comes into play when we need to find the components of a force vector. It's a branch of mathematics that deals with the relationships between the sides and angles of triangles. In our exercise, we use trigonometric functions like sine and cosine to split the forces into x and y components.

• Cosine ( cos ) is used for the horizontal component, as it measures the adjacent side of a right-angle triangle relative to the angle.
• Sine ( sin ) is used for the vertical component, measuring the opposite side relative to the angle.

These trigonometric relationships are key to converting the given force magnitudes and angles into vectors that can be added or subtracted as needed, as we saw in the solution steps.

Force Components

Force components are crucial for understanding how different forces affect a system. By breaking a force into its x and y components, we can more easily analyze and combine forces. In the exercise, each force is split into these components using trigonometric functions.

For example:

• For a force at an angle, the x-component is found using the cosine of the angle.
• The y-component is found using the sine of the angle.

Once each force is broken into these components, the total effect of all the forces can be easily evaluated by simply adding up the respective x and y components. This principle is what allows us to calculate the exact counterforce needed to reach static equilibrium in this exercise.

Newton's First Law

Newton's First Law of Motion, often called the law of inertia, states that an object at rest will stay at rest unless acted upon by a net external force. This concept directly relates to static equilibrium, the main focus of the exercise. In static equilibrium, all forces acting on an object balance each other.

This means the sum of all forces in both the x and y directions must equal zero. In the exercise, by balancing the forces, we see how a fourth force ensures that the sum total is zero, maintaining the system stationary. This profound principle helps us understand why balance is crucial to achieving and maintaining equilibrium.