An ice skater with an initial moment of inertia of 1.44 kg*m^2 is spinning at a rate of 3 revolutions per second. As she extends her arm outward her moment of inertia increases to 2.52 kg*m^2. What is her new angular velocity (in rad/s)?
To find the new angular velocity, we can use the conservation of angular momentum.
The angular momentum (L) is given by the product of the moment of inertia (I) and the angular velocity (ω):
L = I * ω
According to the conservation law, the angular momentum should remain constant before and after extending the arm.
Initially, the angular momentum is:
L1 = I1 * ω1
After extending the arm, the angular momentum is:
L2 = I2 * ω2
Since the angular momentum is conserved, we can equate the two expressions for angular momentum:
L1 = L2
I1 * ω1 = I2 * ω2
We can rearrange this equation to solve for ω2:
ω2 = (I1 * ω1) / I2
Substituting the given values into the equation:
I1 = 1.44 kg*m^2 (initial moment of inertia)
ω1 = 3 revolutions per second * (2π radians per revolution)
I2 = 2.52 kg*m^2 (new moment of inertia)
Converting revolutions per second to radians per second:
ω1 = 3 revolutions per second * (2π radians per revolution) = 6π radians per second
Now, we can solve for ω2:
ω2 = (1.44 kg*m^2 * 6π radians per second) / 2.52 kg*m^2
Simplifying the equation:
ω2 = 8π radians per second
Therefore, the new angular velocity is 8π radians per second.
To solve this problem, we need to use the principle of conservation of angular momentum. According to this principle, the angular momentum of an object remains constant unless an external torque is applied.
The formula for angular momentum is L = Iω, where L is the angular momentum, I is the moment of inertia, and ω is the angular velocity.
Initially, the ice skater has an initial angular momentum, L₁ = I₁ω₁, where I₁ is the initial moment of inertia, and ω₁ is the initial angular velocity.
When she extends her arm, her moment of inertia increases to I₂, and we need to find the new angular velocity ω₂.
Using the conservation of angular momentum, we can equate the initial and final angular momenta:
L₁ = L₂
I₁ω₁ = I₂ω₂
Substituting the given values:
1.44 kg*m^2 * ω₁ = 2.52 kg*m^2 * ω₂
Simplifying the equation, we can divide both sides by 1.44 kg*m^2 to solve for ω₂:
ω₂ = (1.44 kg*m^2 * ω₁) / 2.52 kg*m^2
Therefore, the new angular velocity (ω₂) in rad/s is equal to (1.44 kg*m^2 * ω₁) / 2.52 kg*m^2.