There are two forces on the 2.36 kg box in the overhead view of Fig. 5-31 but only one is shown. For F1 = 18.6 N, a = 11.4 m/s2, and θ = 26.7°, find the second force (a) in unit-vector notation and as (b) a magnitude and (c) a direction. (State the direction as a negative angle measured from the +x direction.)

Question

There are two forces on the 2.36 kg box in the overhead view of Fig. 5-31 but only one is shown. For F1 = 18.6 N, a = 11.4 m/s2, and θ = 26.7°, find the second force (a) in unit-vector notation and as (b) a magnitude and (c) a direction. (State the direction as a negative angle measured from the +x direction.)

Answer 1

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Answer 2

To solve this problem, we need to break down the given information and use vector components to find the second force. Let's go step by step:

Step 1: Draw a diagram
Draw a diagram of the situation described in the problem. Label the forces and angles as mentioned in the question.

Step 2: Resolve the forces into components
The first force F1 can be resolved into its x and y components as follows:
F1x = F1 * cosθ
F1y = F1 * sinθ

Step 3: Calculate the second force
We know that the total force exerted on the box is the sum of the two forces. Therefore, we can write:
F2x + F1x = 0
F2y + F1y = m * a

Substituting the known values:
F2x + F1 * cosθ = 0 ----(1)
F2y + F1 * sinθ = m * a ----(2)

Step 4: Solve for the unknowns
From equation (1), solve for F2x:
F2x = -F1 * cosθ

From equation (2), solve for F2y:
F2y = m * a - F1 * sinθ

Step 5: Write the answer in unit-vector notation
To express the result in unit-vector notation, we divide the forces by their magnitudes to convert them into unit vectors. The unit vector in the x-direction is î, and the unit vector in the y-direction is ĵ.

Therefore, F2 = F2x * î + F2y * ĵ
Substituting the values:
F2 = (-F1 * cosθ) * î + (m * a - F1 * sinθ) * ĵ

Step 6: Calculate the magnitude of the second force
The magnitude of the second force F2 can be found using the equation:
|F2| = √(F2x^2 + F2y^2)

Substituting the values:
|F2| = √((-F1 * cosθ)^2 + (m * a - F1 * sinθ)^2)

Step 7: Calculate the direction of the second force
The direction of the second force can be found using the equation:
θ = atan(F2y/F2x)

Substituting the values and calculating the negative angle measured from the +x direction will give us the direction of the force.

By following these steps and substituting the given values, you will be able to find the second force in both unit-vector notation and as a magnitude and direction.