Final answer:
The energy needed to make the suspension bridge oscillate with an amplitude of 0.100 m is 5.00 × 10^6 J. The time it takes for the bridge's oscillations to go from 0.100 m to 0.500 m amplitude is not affected by the energy imparted by the soldiers marching.
Step-by-step explanation:
(a) Energy calculation:
The energy needed to make the suspension bridge oscillate with an amplitude of 0.100 m can be calculated using the formula for the potential energy of a spring:
Potential Energy = (1/2) * k * A^2
where k is the effective force constant (1.00 × 10^8 N/m) and A is the amplitude (0.100 m).
Plugging in the values:
Potential Energy = (1/2) * (1.00 × 10^8 N/m) * (0.100 m)^2
Potential Energy = 5.00 × 10^6 J
Therefore, the energy needed to make the suspension bridge oscillate with an amplitude of 0.100 m is 5.00 × 10^6 J.
(b) Time calculation:
To calculate the time it takes for the bridge's oscillations to go from 0.100 m to 0.500 m amplitude, we can use the formula for the period of oscillation:
Period = 2π * √(m/k)
where m is the mass of the bridge (not given) and k is the effective force constant (1.00 × 10^8 N/m).
Since the effective force constant and natural frequency are the same, soldiers marching across the bridge with a cadence equal to the bridge's natural frequency and imparting 1.00 × 10^4 J of energy each second would not change the time it takes for the bridge's oscillations to change amplitude.
Therefore, the time it takes for the bridge's oscillations to go from 0.100 m to 0.500 m amplitude is not affected by the energy imparted by the soldiers marching.
Answer Summary:
The energy needed to make the suspension bridge oscillate with an amplitude of 0.100 m is 5.00 × 10^6 J. The time it takes for the bridge's oscillations to go from 0.100 m to 0.500 m amplitude is not affected by the energy imparted by the soldiers marching.