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Chapter 10: Problem 59

A pendulum (mass \(m\), unknown length) moves according to \(x=A\) sin $\omega t .\( (a) Write the equation for \)v_{x}(t)$ and sketch one cycle of the \(v_{x}(t)\) graph. (b) What is the maximum kinetic energy?

Short Answer

Expert verified
Answer: The maximum kinetic energy of the pendulum is \(\frac{1}{2}m(A^2\omega^2)\).

Step by step solution

01

Differentiate position function with respect to time

To find the velocity function \(v_x(t)\), we need to differentiate the given position function \(x(t)\) with respect to time \(t\). Given \(x = A\sin(\omega t)\), we have: \(v_x(t) = \frac{dx}{dt} = \frac{d}{dt}(A\sin(\omega t))\) Using the chain rule, we get: \(v_x(t) = A\omega\cos(\omega t)\)
02

Sketch the graph of \(v_x(t)\) for one cycle

To sketch a graph of \(v_x(t) = A\omega\cos(\omega t)\) for one cycle, we first need to determine the period within which the function completes one full cycle. Since \(\cos(\omega t)\) completes one cycle in a period of \(T = \frac{2\pi}{\omega}\), the graph of \(v_x(t)\) will also complete one cycle in the same period. The graph of the velocity function will be a cosine function with its amplitude equal to \(A\omega\). It starts at a maximum value of \(A\omega\), decreases to \(0\) at \(t = \frac{T}{4}\), goes to the minimum value of \(-A\omega\) at \(t = \frac{T}{2}\), increases to \(0\) at \(t = \frac{3T}{4}\), and finally goes back to the maximum value of \(A\omega\) at \(t=T\).
03

Calculate the maximum kinetic energy

The maximum kinetic energy of the pendulum occurs when the velocity is at its maximum value, which is \(A\omega\). Therefore, the maximum velocity is given by \(v_{max} = A\omega\). Now, we can calculate the max kinetic energy using the formula \(K = \frac{1}{2}mv^2\): \(K_{max} = \frac{1}{2}m(A\omega)^2\) \(K_{max} = \frac{1}{2}m(A^2\omega^2)\) Therefore, the maximum kinetic energy of the pendulum is \(\frac{1}{2}m(A^2\omega^2)\).

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Most popular questions from this chapter

(a) What is the energy of a pendulum \((L=1.0 \mathrm{m}, m=\) $0.50 \mathrm{kg})\( oscillating with an amplitude of \)5.0 \mathrm{cm} ?$ (b) The pendulum's energy loss (due to damping) is replaced in a clock by allowing a \(2.0-\mathrm{kg}\) mass to drop \(1.0 \mathrm{m}\) in 1 week. What average percentage of the pendulum's energy is lost during one cycle?
The ratio of the tensile (or compressive) strength to the density of a material is a measure of how strong the material is "pound for pound." (a) Compare tendon (tensile strength \(80.0 \mathrm{MPa}\), density $1100 \mathrm{kg} / \mathrm{m}^{3}\( ) with steel (tensile strength \)\left.0.50 \mathrm{GPa}, \text { density } 7700 \mathrm{kg} / \mathrm{m}^{3}\right)$ which is stronger "pound for pound" under tension? (b) Compare bone (compressive strength \(160 \mathrm{MPa}\), density $1600 \mathrm{kg} / \mathrm{m}^{3}$ ) with concrete (compressive strength $\left.0.40 \mathrm{GPa}, \text { density } 2700 \mathrm{kg} / \mathrm{m}^{3}\right):$ which is stronger "pound for pound" under compression?
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