Scelta multipla
Which of the following best describes the relationship between the speed of an object and its kinetic energy?
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9. Work & Energy
Intro to Energy & Kinetic Energy
Scelta multipla
A block attached to an ideal spring oscillates with amplitude and total mechanical energy . If the total energy is doubled to , what will the new amplitude of oscillation be?
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Recall that the total mechanical energy \(E\) of a block-spring system undergoing simple harmonic motion is given by the formula \(E = \frac{1}{2} k A^2\), where \(k\) is the spring constant and \(A\) is the amplitude of oscillation.
Since the problem states that the total energy is doubled from \(E\) to \$2E$, set up the equation for the new energy: \(2E = \frac{1}{2} k A_{new}^2\), where $A_{new}$ is the new amplitude.
Substitute the original energy expression \(E = \frac{1}{2} k A^2\) into the doubled energy equation to get \(2 \times \frac{1}{2} k A^2 = \frac{1}{2} k A_{new}^2\).
Simplify the equation to isolate \(A_{new}^2\): \(k A^2 = \frac{1}{2} k A_{new}^2\), then divide both sides by \(k\) to get \(A^2 = \frac{1}{2} A_{new}^2\).
Solve for \(A_{new}\) by multiplying both sides by 2 and taking the square root: \(A_{new} = \sqrt{2} A\). This shows that the new amplitude is \(\sqrt{2}\) times the original amplitude.
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