A simple pendulum of length 1.00 m has a mass of 100 g attached. It is drawn back 30.0° and then released. What is the maximum speed of the mass?
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Answer: The maximum speed of the mass is approximately 1.72 m/s.
The motion of a simple pendulum is governed by the equation:
T = 2π √(L/g)
where T is the period of the pendulum, L is its length, and g is the acceleration due to gravity (9.81 m/s^2 near the Earth's surface).
The maximum speed of the mass occurs at the bottom of its swing, where all of the potential energy has been converted to kinetic energy. At this point, the velocity of the mass can be calculated using the conservation of energy:
PE = KE
mgh = (1/2)mv^2
where m is the mass of the pendulum, h is the height of the pendulum above its lowest point, and v is the velocity of the pendulum at its lowest point.
In this case, the height of the pendulum above its lowest point is L(1 - cosθ), where θ is the initial angle of the pendulum. Thus, we have:
h = L(1 - cosθ) = 1.00 m(1 - cos30°) = 0.134 m
Substituting this value into the conservation of energy equation, we have:
(0.100 kg)(9.81 m/s^2)(0.134 m) = (1/2)(0.100 kg)v^2
Simplifying and solving for v, we get:
v = √(2gh) = √(2(9.81 m/s^2)(0.134 m)) ≈ 1.72 m/s
Therefore, the maximum speed of the mass is approximately 1.72 m/s.
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