Harmonic motion of a trampoline
A 28 kg child gets on a trampoline, which may be considered a massless vertical spring, depressing it 23 cm. The child begins to bounces, causing the trampoline to oscillate vertically with simple harmonic motion. What is the period of oscillation and at what amplitude would the child begin to be airborne?
Answer
Like the problem says, you can treat the trampoline as a vertical spring with a mass attached to the end (the child). You can find the spring constant by setting the spring force equal to the force of gravity on the child. From here, you can use the equations you know to find the angular frequency and the period.
To find when the child goes airborne, first consider the net force on the child when it is in contact with the trampoline. You know the position is a simple sine or cosine with time, so you can differentiate twice to find the acceleration. Multiply by the child's mass and you know the net force that must be on the child.
When the child goes airborne, right when they leave contact with the trampoline, the net force on the child is just the force of gravity. In other words, the maximum amplitude the child could attain before going airborne is when the force of gravity equals the maximum of the force you found above. From here, try solving for the amplitude.
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Like the problem says, you can treat the trampoline as a vertical spring with a mass attached to the end (the child). You can find the spring constant by setting the spring force equal to the force of gravity on the child. From here, you can use the equations you know to find the angular frequency and the period.
To find when the child goes airborne, first consider the net force on the child when it is in contact with the trampoline. You know the position is a simple sine or cosine with time, so you can differentiate twice to find the acceleration. Multiply by the child's mass and you know the net force that must be on the child.
When the child goes airborne, right when they leave contact with the trampoline, the net force on the child is just the force of gravity. In other words, the maximum amplitude the child could attain before going airborne is when the force of gravity equals the maximum of the force you found above. From here, try solving for the amplitude.