The figure shows an overhead view of a 0.029 kg lemon half and two of the three horizontal forces that act on it as it is on a frictionless table. Force ₁ has a magnitude of 8 N and is at 0₁ = 27°. Force F2 has a magnitude of 10 N and is at 02 = 30°. In unit-vector notation, what is the third force if the lemon half (a) is stationary, (b) has the constant velocity = (12î - 123) m/s, and (c) has the v = (14tî - 15tĵ) m/s², where t is time? 0₁7 162 x

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Chapter1: Units, Trigonometry. And Vectors
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### Problem Statement

The figure shows an overhead view of a 0.029 kg lemon half and two of the three horizontal forces that act on it as it is on a frictionless table. 

- Force \( \vec{F}_1 \) has a magnitude of 8 N and is at \( \theta_1 = 27^\circ \).
- Force \( \vec{F}_2 \) has a magnitude of 10 N and is at \( \theta_2 = 30^\circ \).

In unit-vector notation, what is the third force if the lemon half:

(a) is stationary,

(b) has the constant velocity \( \vec{v} = (12\hat{i} - 12\hat{j}) \) m/s,

(c) has the velocity \( \vec{v} = (14t\hat{i} - 15t\hat{j}) \) m/s², where t is time?

### Diagram Description

The diagram features a yellow circle representing the lemon half at the origin of an x-y coordinate system with two forces acting on it. 

- The first force \( \vec{F}_1 \) is represented by a vector pointing upwards and slightly to the left, forming an angle \( \theta_1 \) of 27 degrees with the positive x-axis.
- The second force \( \vec{F}_2 \) is represented by a vector pointing downwards and to the right, forming an angle \( \theta_2 \) of 30 degrees with the positive x-axis.

### Input Fields for Solutions

**(a)** Force \( \vec{F}_3 \) if the lemon half is stationary:
- \(  \text{Number} \) i + \( \text{Number} \) j \( \text{Units} \)

**(b)** Force \( \vec{F}_3 \) if the lemon half has constant velocity \( \vec{v} = (12\hat{i} - 12\hat{j}) \) m/s:
- \(  \text{Number} \) i + \( \text{Number} \) j \( \text{Units} \)

**(c)** Force \( \vec{F}_3 \) if the lemon half has a velocity \( \vec{v} = (14t\hat{i} - 15t\hat{j}) \) m/s
Transcribed Image Text:### Problem Statement The figure shows an overhead view of a 0.029 kg lemon half and two of the three horizontal forces that act on it as it is on a frictionless table. - Force \( \vec{F}_1 \) has a magnitude of 8 N and is at \( \theta_1 = 27^\circ \). - Force \( \vec{F}_2 \) has a magnitude of 10 N and is at \( \theta_2 = 30^\circ \). In unit-vector notation, what is the third force if the lemon half: (a) is stationary, (b) has the constant velocity \( \vec{v} = (12\hat{i} - 12\hat{j}) \) m/s, (c) has the velocity \( \vec{v} = (14t\hat{i} - 15t\hat{j}) \) m/s², where t is time? ### Diagram Description The diagram features a yellow circle representing the lemon half at the origin of an x-y coordinate system with two forces acting on it. - The first force \( \vec{F}_1 \) is represented by a vector pointing upwards and slightly to the left, forming an angle \( \theta_1 \) of 27 degrees with the positive x-axis. - The second force \( \vec{F}_2 \) is represented by a vector pointing downwards and to the right, forming an angle \( \theta_2 \) of 30 degrees with the positive x-axis. ### Input Fields for Solutions **(a)** Force \( \vec{F}_3 \) if the lemon half is stationary: - \( \text{Number} \) i + \( \text{Number} \) j \( \text{Units} \) **(b)** Force \( \vec{F}_3 \) if the lemon half has constant velocity \( \vec{v} = (12\hat{i} - 12\hat{j}) \) m/s: - \( \text{Number} \) i + \( \text{Number} \) j \( \text{Units} \) **(c)** Force \( \vec{F}_3 \) if the lemon half has a velocity \( \vec{v} = (14t\hat{i} - 15t\hat{j}) \) m/s
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