### Problem ExplanationA crate sits on a wooden horizontal surface. To move it in the x-direction, Sam and Tom apply forces \( F_1 \) and \( F_2 \) at angles \( \theta_1 \) and \( \theta_2 \), respectively. The resultant force \( F_r = 28.5 \, \text{lbs} \) in the x-direction is required. All forces are in the xy plane.Sam applies a force of \( F_1 = 19.5 \, \text{lbs} \) at an angle of \( \theta_1 = 16^\circ \) from the positive x-axis. The task is to find the magnitude and angle of the force \( F_2 \) that Tom must apply.#### Diagram ExplanationThe diagram is a top-down view of a crate being pushed. It shows the directions of the forces \( F_1 \) and \( F_2 \), both making angles \( \theta_1 \) and \( \theta_2 \) with respect to the x-axis.### Questions and Equations1. **Expression for \( F_1 \):** Write an expression for \( F_1 \) as the crate starts moving using the sum of the forces in the x-direction: \[ F_r = F_1 \cos(\theta_1) + F_2 \cos(\theta_2) \]2. **Sum of Forces in the y-direction:** Write an equation for the sum of forces in the y-direction just as the crate starts to move: \[ 0 = F_1 \sin(\theta_1) - F_2 \sin(\theta_2) \]3. **Finding \( \theta_2 \):** Combine these equations to develop an expression for \( \tan(\theta_2) \): \[ \tan(\theta_2) = \frac{F_1 \sin(\theta_1)}{F_r - F_1 \cos(\theta_1)} \] Solve for the value of \( \theta_2 \) in degrees.### Calculation Steps1. Use the given values to plug into the equations.2. Solve the equations to find the magnitude of \( F_2 \) and angle \( \theta_2 \).These steps will achieve

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A crate sits on a wooden horizontal surface, Sam and Tom apply forces F₁ and F₂ at angles 0₁ and 0₂ respectively with the goal of moving the crate in the x-direction. A resultant force F, = 28.5 lbs in the x - direction is required to accomplish this. All the forces are in the xy plane. If Sam applies a force of F₁ = 19.5 lbs at an angle of 0₁= 16° from the positive x-axis, complete the following steps to determine the magnitude and angle of force that Tom must apply. (The drawing shows a top down view of looking down onto the crate). Lx m Write an expression for F,, just as the crate starts to move using the sum of the forces in the x-direction in terms of the variables F₁, F₂, 0₁, and 0₂. Write an equation for the sum of forces in the y-direction when the crate just starts to move using the coordinate system and in terms of the variables F₁, F₂, 0₁, and 0₂. Combine these two equations to develop an expression for tan (0₂) in terms of F, F₁, F₂, 0, and solve for the value of 0₂ in degrees.

A crate sits on a wooden horizontal surface, Sam and Tom apply forces F₁ and F₂ at angles 0₁ and 0₂ respectively with the goal of moving the crate in the x-direction. A resultant force F, = 28.5 lbs in the x - direction is required to accomplish this. All the forces are in the xy plane. If Sam applies a force of F₁ = 19.5 lbs at an angle of 0₁= 16° from the positive x-axis, complete the following steps to determine the magnitude and angle of force that Tom must apply. (The drawing shows a top down view of looking down onto the crate). Lx m Write an expression for F,, just as the crate starts to move using the sum of the forces in the x-direction in terms of the variables F₁, F₂, 0₁, and 0₂. Write an equation for the sum of forces in the y-direction when the crate just starts to move using the coordinate system and in terms of the variables F₁, F₂, 0₁, and 0₂. Combine these two equations to develop an expression for tan (0₂) in terms of F, F₁, F₂, 0, and solve for the value of 0₂ in degrees.

Physics for Scientists and Engineers: Foundations and Connections

1st Edition

ISBN:9781133939146

Author:Katz, Debora M.

Publisher:Katz, Debora M.

Chapter9: Energy In Nonisolated Systems

Section: Chapter Questions

Problem 15PQ

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### Problem Explanation

A crate sits on a wooden horizontal surface. To move it in the x-direction, Sam and Tom apply forces \( F_1 \) and \( F_2 \) at angles \( \theta_1 \) and \( \theta_2 \), respectively. The resultant force \( F_r = 28.5 \, \text{lbs} \) in the x-direction is required. All forces are in the xy plane.

Sam applies a force of \( F_1 = 19.5 \, \text{lbs} \) at an angle of \( \theta_1 = 16^\circ \) from the positive x-axis. The task is to find the magnitude and angle of the force \( F_2 \) that Tom must apply.

#### Diagram Explanation

The diagram is a top-down view of a crate being pushed. It shows the directions of the forces \( F_1 \) and \( F_2 \), both making angles \( \theta_1 \) and \( \theta_2 \) with respect to the x-axis.

### Questions and Equations

1. **Expression for \( F_1 \):**

Write an expression for \( F_1 \) as the crate starts moving using the sum of the forces in the x-direction:

\[
F_r = F_1 \cos(\theta_1) + F_2 \cos(\theta_2)
\]

2. **Sum of Forces in the y-direction:**

Write an equation for the sum of forces in the y-direction just as the crate starts to move:

\[
0 = F_1 \sin(\theta_1) - F_2 \sin(\theta_2)
\]

3. **Finding \( \theta_2 \):**

Combine these equations to develop an expression for \( \tan(\theta_2) \):

\[
\tan(\theta_2) = \frac{F_1 \sin(\theta_1)}{F_r - F_1 \cos(\theta_1)}
\]

Solve for the value of \( \theta_2 \) in degrees.

### Calculation Steps

1. Use the given values to plug into the equations.
2. Solve the equations to find the magnitude of \( F_2 \) and angle \( \theta_2 \).

These steps will achieve

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