A $4.00 \mathrm{~kg}$ mass is connected to a spring with a spring constant of $9.00 \mathrm{~N} / \mathrm{m}$. The velocity is given by the expression $v(t)=(12.8 \mathrm{~cm} / \mathrm{s}) \cos (\omega t+\pi / 4)$. What is the maximum acceleration of the mass?
A) $13.6 \mathrm{~cm} / \mathrm{s}^{2}$
B) $20.2 \mathrm{~cm} / \mathrm{s}^{2}$
C) $19.2 \mathrm{~cm} / \mathrm{s}^{2}$
D) $24.5 \mathrm{~cm} / \mathrm{s}^{2}$
E) $16.2 \mathrm{~cm} / \mathrm{s}^{2}$

Question

Quizplus · Accepted Answer

The maximum acceleration $a_{\text{max}}$ can be found using the formula $a_{\text{max}} = \omega^2 A$, where $A$ is the amplitude of the motion and $\omega$ is the angular frequency. Given $v(t)=(12.8 \mathrm{~cm} / \mathrm{s}) \cos (\omega t+\pi / 4)$, the amplitude $A = 12.8 \mathrm{~cm}$. The angular frequency $\omega = \sqrt{\frac{k}{m}}$, where $k = 9.00 \mathrm{~N/m}$ and $m = 4.00 \mathrm{~kg}$. Converting units, $k = 9.00 \mathrm{~N/m} = 9.00 \mathrm{~kg/s^2}$ (since $1 \mathrm{~N/m} = 1 \mathrm{~kg/s^2}$). Thus, $\omega = \sqrt{\frac{9.00}{4.00}} = \frac{3}{2}$ rad/s. The maximum acceleration is then $a_{\text{max}} = \left(\frac{3}{2}\right)^2 \times 12.8 \mathrm{~cm/s^2} = 2.25 \times 12.8 \mathrm{~cm/s^2} = 28.8 \mathrm{~cm/s^2}$. However, this calculation error led to an incorrect final step. Correctly, $a_{\text{max}} = \omega^2 A = \left(\frac{3}{2}\right)^2 \times 12.8 = 2.25 \times 12.8 = 28.8 \mathrm{~cm/s^2}$, but the correct approach should yield $a_{\text{max}} = \omega^2 \times A$, with $\omega$ correctly calculated from $\omega = \sqrt{\frac{k}{m}}$, and ensuring units are consistent for $A$. The mistake in the final calculation step led to an incorrect conclusion. Correctly calculating, with $\omega = \sqrt{\frac{9}{4}} = 1.5 \, \text{s}^{-1}$, and $A = 12.8 \, \text{cm} = 0.128 \, \text{m}$, we should find $a_{\text{max}}$ with correct unit conversion and calculation.

A 4.00 kg4.00 \mathrm{~kg}4.00 kg Mass Is Connected to a Spring with a Spring Constant

A $4.00 \mathrm{~kg}$ Mass Is Connected to a Spring with a Spring Constant