a woman stands at the center of a platform. the woman and the platform rotate with an angular speed of 5.00 rad/s. friction is negligible. her arms are outstretched, and she is holding a dumbbell in each hand. in this position the total moment of inertia of the rotating system is 5.40 kgm^2. by pulling in her arms, the moment of inertia is reduced to 3.80 kg m^2. find her new angular speed.
conservation of momentum applies
I*w=I'*w'
w'=(5.4/3.9)5 rad/sec
To find the new angular speed, we can use the principle of conservation of angular momentum. According to this principle, the initial angular momentum of the system must be equal to the final angular momentum.
The angular momentum (L) of an object is calculated by multiplying its moment of inertia (I) by its angular speed (ω):
L = I * ω
The initial angular momentum (L1) is equal to the final angular momentum (L2). We can write this as:
L1 = L2
The initial moment of inertia (I1) is 5.40 kgm^2, and the initial angular speed (ω1) is 5.00 rad/s. The final moment of inertia (I2) is 3.80 kg m^2, and we need to find the final angular speed (ω2).
Using the equation above, we can set up the equation:
I1 * ω1 = I2 * ω2
Plug in the values:
(5.40 kgm^2) * (5.00 rad/s) = (3.80 kg m^2) * ω2
Solving for ω2:
ω2 = (5.40 kgm^2 * 5.00 rad/s) / (3.80 kg m^2)
ω2 = 7.11 rad/s (rounded to two decimal places)
Therefore, her new angular speed is approximately 7.11 rad/s.
To find the new angular speed, we can use the principle of conservation of angular momentum. According to this principle, the initial angular momentum of the system is equal to the final angular momentum of the system.
The angular momentum of the system with the outstretched arms can be calculated using the formula:
L1 = I1 * ω1
Where:
L1 is the initial angular momentum
I1 is the initial moment of inertia (5.40 kg m^2)
ω1 is the initial angular speed (5.00 rad/s)
Similarly, the angular momentum of the system with the arms pulled in can be calculated using the formula:
L2 = I2 * ω2
Where:
L2 is the final angular momentum
I2 is the final moment of inertia (3.80 kg m^2)
ω2 is the final angular speed (unknown)
Since angular momentum is conserved, we have:
L1 = L2
Substituting the values, we get:
I1 * ω1 = I2 * ω2
Solving for ω2, we get:
ω2 = (I1 * ω1) / I2
Substituting the given values:
ω2 = (5.40 kg m^2 * 5.00 rad/s) / 3.80 kg m^2
Now, let's calculate the value:
ω2 = 7.1053 rad/s (rounded to five decimal places)
Therefore, the woman's new angular speed is approximately 7.1053 rad/s.