A constant force of 10 N pushes a particle along the x-axis. The position of the particle is represented by x=11m-(2m/s)t+(0.5m/s^2)t^2. Find the work done by the force between t=0... A constant force of 10 N pushes a particle along the x-axis. The position of the particle is represented by x=11m-(2m/s)t+(0.5m/s^2)t^2. Find the work done by the force between t=0s and t=1s and between t=1s and t=2s. Is the force conservative?
Understand the Problem
The question requires us to calculate the work done by a constant force of 10 N on a particle whose position is given as a function of time. We need to find the work done in two time intervals: from t=0s to t=1s and from t=1s to t=2s. Additionally, we must determine if the force is conservative.
Answer
$W_1 = -15 \text{ J}, W_2 = -5 \text{ J}; \text{ Force is conservative}$.
Answer for screen readers
The work done by the force from $t = 0s$ to $t = 1s$ is $W_1 = -15 \text{ J}$, and from $t = 1s$ to $t = 2s$ is $W_2 = -5 \text{ J}$. The force is conservative.
Steps to Solve
- Identify the Position Function
The position of the particle is given by the equation: $$ x(t) = 11 - 2t + 0.5t^2 $$
- Calculate Positions at Time Intervals
Determine the position of the particle at $t = 0s$, $t = 1s$, and $t = 2s$.
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For $t = 0s$: $$ x(0) = 11 - 2(0) + 0.5(0)^2 = 11 \text{ m} $$
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For $t = 1s$: $$ x(1) = 11 - 2(1) + 0.5(1)^2 = 11 - 2 + 0.5 = 9.5 \text{ m} $$
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For $t = 2s$: $$ x(2) = 11 - 2(2) + 0.5(2)^2 = 11 - 4 + 2 = 9 \text{ m} $$
- Calculate Displacement
Find the displacement for both time intervals:
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From $t = 0s$ to $t = 1s$: $$ \Delta x_1 = x(1) - x(0) = 9.5 - 11 = -1.5 \text{ m} $$
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From $t = 1s$ to $t = 2s$: $$ \Delta x_2 = x(2) - x(1) = 9 - 9.5 = -0.5 \text{ m} $$
- Calculate Work Done
Use the work formula: $$ W = F \cdot \Delta x $$ where the force $F = 10 \text{ N}$.
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Work done from $t = 0s$ to $t = 1s$: $$ W_1 = F \cdot \Delta x_1 = 10 \cdot (-1.5) = -15 \text{ J} $$
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Work done from $t = 1s$ to $t = 2s$: $$ W_2 = F \cdot \Delta x_2 = 10 \cdot (-0.5) = -5 \text{ J} $$
- Determine if the Force is Conservative
A force is conservative if the work done is independent of the path taken. Since the force is constant and we only consider translational movement along the x-axis, the force can be classified as conservative.
The work done by the force from $t = 0s$ to $t = 1s$ is $W_1 = -15 \text{ J}$, and from $t = 1s$ to $t = 2s$ is $W_2 = -5 \text{ J}$. The force is conservative.
More Information
In physics, work done by a constant force can vary based on the displacement, and it represents energy transferred to or from an object. Conservative forces are crucial in understanding energy conservation in mechanics.
Tips
- Confusing position with displacement; always calculate the change in position ($\Delta x$) for the correct interval.
- Not considering the direction of the force and displacement; work can be negative if the force opposes the motion.
- Forgetting to apply the work formula correctly.
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