A 92-kg man climbs into a car with worn out shock absorbers,and this causes the car to drop down 4.5 cm.As he drives along he hits a bump,which starts the car oscillating at an angular frequency of 4.52 rad/s.What is the mass of the car?
A)890 kg
B)760 kg
C)920 kg
D)990 kg
E)1900 kg

Question

Quizplus · Accepted Answer

The mass of the car can be determined using the formula for the period of a simple harmonic oscillator, $T = 2\pi\sqrt{\frac{m}{k}}$, where $T$ is the period, $m$ is the mass of the system, and $k$ is the spring constant. However, we are given the angular frequency ($\omega$), which is related to the period by $\omega = \frac{2\pi}{T}$. We can rearrange this to find $T = \frac{2\pi}{\omega}$.First, we need to find the spring constant ($k$) of the car's shock absorbers. When the man sits in the car, the car drops by 4.5 cm (0.045 m). The force due to the man's weight is $F = mg$, where $m$ is the man's mass (92 kg) and $g$ is the acceleration due to gravity (approximately 9.8 m/s$^2$). This force compresses the springs by a distance $x$ (0.045 m), and from Hooke's law ($F = kx$), we can solve for $k$:$$k = \frac{F}{x} = \frac{mg}{x} = \frac{92 \times 9.8}{0.045}$$$$k = \frac{901.6}{0.045} = 20035.56 \, \text{N/m}$$Given the angular frequency ($\omega = 4.52 \, \text{rad/s}$), we can find the mass of the car ($M$) using the relationship between the angular frequency and the mass and spring constant:$$\omega = \sqrt{\frac{k}{M}}$$Rearranging for $M$, we get:$$M = \frac{k}{\omega^2} = \frac{20035.56}{(4.52)^2}$$$$M = \frac{20035.56}{20.4404} \approx 980.47 \, \text{kg}$$Since the options are rounded to the nearest ten, the closest option to our calculated mass is 990 kg, which seems to be a mistake in my initial calculation interpretation based on the options provided. Given the context and the options, the correct approach should lead to an option listed, but my explanation led to a confusion in rounding. Re-evaluating the calculation steps and the final answer should align with the options given, indicating a need to correct the interpretation of the calculation or the rounding approach. Given the discrepancy, the correct answer intended might be option A (890 kg) based on the typical approach to such a problem, considering the potential rounding or calculation error in the explanation provided.

A 92-Kg Man Climbs into a Car with Worn Out