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The position vector of a particle is given by r = 11ti + 2.0t²j - 0.9(t3-5)k, where t is the time in seconds from the start of the motion and where r is expressed in meters. For the condition when t = 4 s, determine the power P developed by the force F = 36i - 21j - 46k N which acts on the particle. Answer: P = i kW

The position vector of a particle is given by r = 11ti + 2.0t²j - 0.9(t3-5)k, where t is the time in seconds from the start of the motion and where r is expressed in meters. For the condition when t = 4 s, determine the power P developed by the force F = 36i - 21j - 46k N which acts on the particle. Answer: P = i kW

Elements Of Electromagnetics
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
Section: Chapter Questions
Problem 1.1MA
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### Problem Statement The position vector of a particle is given by: \[ \mathbf{r} = 11t\mathbf{i} + 2.0t^2\mathbf{j} - 0.9(t^3 - 5)\mathbf{k} \] where \( t \) is the time in seconds from the start of the motion and \(\mathbf{r}\) is expressed in meters. For the condition when \( t = 4 \) seconds, determine the power \( P \) developed by the force: \[ \mathbf{F} = 36\mathbf{i} - 21\mathbf{j} - 46\mathbf{k} \, \text{N} \] which acts on the particle. ### Solution Answer: \( P \) = \[ \underline{\makebox[3cm]{}} \] kW ### Explanation To determine the power \( P \) developed by the force on the particle, follow these steps: 1. **Find the velocity vector \(\mathbf{v}\) by differentiating the position vector \(\mathbf{r}\) with respect to time \( t \).** \[ \mathbf{v} = \frac{d\mathbf{r}}{dt} \] 2. **Evaluate the velocity vector \(\mathbf{v}\) at \( t = 4 \) seconds.** 3. **Calculate the instantaneous power \( P \) using the formula:** \[ P = \mathbf{F} \cdot \mathbf{v} \] Here, \( \cdot \) represents the dot product of the force vector and the velocity vector. 4. **Convert the power to kilowatts (kW) if necessary.**
Transcribed Image Text:### Problem Statement The position vector of a particle is given by: \[ \mathbf{r} = 11t\mathbf{i} + 2.0t^2\mathbf{j} - 0.9(t^3 - 5)\mathbf{k} \] where \( t \) is the time in seconds from the start of the motion and \(\mathbf{r}\) is expressed in meters. For the condition when \( t = 4 \) seconds, determine the power \( P \) developed by the force: \[ \mathbf{F} = 36\mathbf{i} - 21\mathbf{j} - 46\mathbf{k} \, \text{N} \] which acts on the particle. ### Solution Answer: \( P \) = \[ \underline{\makebox[3cm]{}} \] kW ### Explanation To determine the power \( P \) developed by the force on the particle, follow these steps: 1. **Find the velocity vector \(\mathbf{v}\) by differentiating the position vector \(\mathbf{r}\) with respect to time \( t \).** \[ \mathbf{v} = \frac{d\mathbf{r}}{dt} \] 2. **Evaluate the velocity vector \(\mathbf{v}\) at \( t = 4 \) seconds.** 3. **Calculate the instantaneous power \( P \) using the formula:** \[ P = \mathbf{F} \cdot \mathbf{v} \] Here, \( \cdot \) represents the dot product of the force vector and the velocity vector. 4. **Convert the power to kilowatts (kW) if necessary.**
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