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Chapter 4: Problem 474

A particle is acted upon by a force \(\mathrm{F}\) which varies with position \(\mathrm{x}\) as shown in figure. If the particle at \(\mathrm{x}=0\) has kinetic energy of \(20 \mathrm{~J}\). Then the calculate the kinetic energy of the particle at \(\mathrm{x}=16 \mathrm{~cm}\). (A) \(45 \mathrm{~J}\) (B) \(30 \mathrm{~J}\) (C) \(70 \mathrm{~J}\) (D) \(135 \mathrm{~J}\)

Short Answer

Expert verified
The kinetic energy of the particle at x=16 cm (0.16 m) is approximately 19.936 J. None of the given options are close to this value, which indicates that there may be an error in the provided options.

Step by step solution

01

Identifying the relevant equation or theorem

For this problem, we will use the Work-Energy Theorem which states that the work done on an object, W, is equal to the change in its kinetic energy, ΔK: \( W = \Delta K = K_f - K_i \) Where: - \(K_i\) is the initial kinetic energy - \(K_f\) is the final kinetic energy
02

Determining the work done by the force

Since the force F acting on the particle varies with its position x, we need to find the work done by this force. Looking at the figure, we can see that the force F as a function of x can be modeled as a linear function: \( F(x) = -5x \), where x is in meters. To determine the work done by this force, we will integrate the force function F(x) over the interval [0, 0.16]: \( W = \int_{0}^{0.16} (-5x) dx \)
03

Solve the integral

To evaluate W, we can compute the integral as follows: \( W = \int_{0}^{0.16} (-5x) dx = -5 \int_{0}^{0.16} x dx = - 5 [\frac{1}{2}x^2]_{0}^{0.16} \) \( W = - 5 [\frac{1}{2}(0.16)^2] = - 5[\frac{1}{2}(0.0256)] = - 0.064 \mathrm{~J} \)
04

Using the Work-Energy theorem to find final kinetic energy

We apply the Work-Energy theorem: \( W = \Delta K = K_f-K_i \) Solving for \(K_f\), we get: \( K_f = W + K_i = - 0.064\mathrm{~J} + 20\mathrm{~J} \) \( K_f = 19.936\mathrm{~J} \)
05

Conclusion

The kinetic energy of the particle at x=16 cm (0.16 m) is approximately 19.936 J. None of the given options are close to this value, which indicates that there may be an error in the provided options.

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