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Unraveling Bead Motion: Force, Acceleration & Energy
Chapter 7, Problem 4(choose chapter or problem)
A bead with mass \(1.8\times {{10}^{-2}}\,\text{kg}\) is moving along a wire in the positive direction of an x-axis. Beginning at time t=0, when the bead passes through x=0 with speed 12.0 m/s, a constant force acts on the bead. Figure 7-22 indicates the bead's position at these four times: \({{t}_{0}}=0,\ {{t}_{1}}=1.0\ \text{s},\ {{t}_{2}}=2.0\ \text{s}\), and \({{t}_{3}}=3.0\ \text{s}\). The bead momentarily stops at \({{t}_{3}}=3.0\ \text{s}\). What is the kinetic energy of the bead at t = 10 s?
Questions & Answers
QUESTION:
A bead with mass \(1.8\times {{10}^{-2}}\,\text{kg}\) is moving along a wire in the positive direction of an x-axis. Beginning at time t=0, when the bead passes through x=0 with speed 12.0 m/s, a constant force acts on the bead. Figure 7-22 indicates the bead's position at these four times: \({{t}_{0}}=0,\ {{t}_{1}}=1.0\ \text{s},\ {{t}_{2}}=2.0\ \text{s}\), and \({{t}_{3}}=3.0\ \text{s}\). The bead momentarily stops at \({{t}_{3}}=3.0\ \text{s}\). What is the kinetic energy of the bead at t = 10 s?
Step 1 of 3
Consider the given data as follows.
The mass of the bead is \(m=1.8\times {{10}^{-2}}\,\text{kg}\).
The initial speed at x=0 is \(u=12\ \text{m/s}\).
The position of the bead at \(t=1\,\text{s}\) is \(x=10\ \text{m}\).
Now, using the second equation of motion, we have the following.
\(s=ut+\dfrac{1}{2}a{{t}^{2}}\)
By plugging all the values, we get,
\(10\ \text{m}=\left( 12\ \text{m/s}\times 1\,\text{s} \right)+\left( \dfrac{1}{2}\times a\times 1\ \text{s} \right) \)
\( a=2\left( 10\ \text{m}-12\ \text{m/s} \right) \)
\( =-4\ \text{m/}{{\text{s}}^{2}}\)
Here, a negative sign shows the direction of the acceleration.
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Unraveling Bead Motion: Force, Acceleration & Energy
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Explore the dynamics of a bead under constant force. Understand its acceleration, velocity over time, and how to calculate its kinetic energy. Witness the interplay of motion equations and real-world applications in a simple system.