A runner of mass m = 58 kg and running at 2 m/s runs as shown and jumps on the rim of a playground merry-go-round which has a moment of inertia of 408 kg·m2 and a radius of 2 meters. Assuming the merry-go-round is initially at rest, what is its final angular velocity to three decimal places?
initial angular momentum=final ang momentum
Imgo*0+58*2^2*2/2PI2=(Imgo+58*2^2)w
where w is the final angular velocity in rad/sec. Notice 2/2pi2 is the initial angular velocity of the runners mass
To determine the final angular velocity of the merry-go-round, we need to apply the principle of conservation of angular momentum. According to this principle, the total angular momentum before the runner jumps onto the merry-go-round should be equal to the total angular momentum after the event.
The angular momentum of an object can be calculated using the formula:
L = I * ω
where L is the angular momentum, I is the moment of inertia, and ω is the angular velocity.
Initially, the merry-go-round is at rest, so its initial angular momentum is zero:
L_initial = 0
When the runner lands on the rim of the merry-go-round, they contribute to the total angular momentum. The angular momentum of the runner can be calculated as:
L_runner = m * r * v
where m is the mass of the runner, r is the radius of the merry-go-round, and v is the velocity of the runner.
The total angular momentum after the runner jumps onto the merry-go-round will be:
L_total = L_merry-go-round + L_runner
Since L_initial = L_total, we can set up the equation:
0 = I * ω_final + m * r * v
Solving for ω_final, we get:
ω_final = -m * r * v / I
Now we can substitute the given values into the formula to calculate the final angular velocity:
ω_final = -(58 kg)(2 m)(2 m/s) / 408 kg·m^2
Evaluating this expression, we find:
ω_final ≈ -0.568 rad/s
However, since angular velocity is a vector quantity and negative angular velocity means the merry-go-round is spinning in the opposite direction of the runner's motion, we can convert this negative angular velocity to a positive value for simplicity. Thus, the final angular velocity is:
ω_final ≈ |0.568| rad/s ≈ 0.568 rad/s
Therefore, the final angular velocity of the merry-go-round, to three decimal places, is approximately 0.568 rad/s.