Answer:
The time period of the empty car will be "1.00 s".
Explanation:
The given values in the question will be:
Mass,
m = 250 kg
Loaded car's time period will be:
T = 1.08 s
Shock absorbers compression,
x = 4 cm
or,
= 0.04 m
Now,
Weight of passengers will be:
⇒ [tex]F=mg[/tex]
[tex]=250\times 9.8[/tex]
[tex]=2450 \ N[/tex]
The spring constant of shock absorbers will be:
⇒ [tex]k=\frac{F}{x}[/tex]
[tex]=\frac{2450}{0.04}[/tex]
[tex]=61.250 \ N/m[/tex]
As we know,
Time period, [tex]T = 2 \pi\sqrt{\frac{M}{k} }[/tex]
On substituting the values, we get
[tex]1.08=2\pi \sqrt{\frac{M}{61250} }[/tex]
[tex]\frac{M}{61250}=0.02955[/tex]
[tex]M=0.02955\times 61250[/tex]
[tex]=1809.6 \ kg[/tex] (Total mass of car as well as its passengers)
Now,
The mass of the empty car will be:
⇒ [tex]m'=M-m[/tex]
[tex]=1809.6-250[/tex]
[tex]=1559.6 \ kg[/tex]
hence,
The time period of empty car will be:
⇒ [tex]T'=2\pi\sqrt{\frac{m'}{k} }[/tex]
[tex]=2\pi\sqrt{\frac{1559.6}{61250} }[/tex]
[tex]=2\pi \sqrt{0.0254}[/tex]
[tex]=1.003 \ s[/tex]
or,
[tex]=1.00 \ s[/tex]
