A wheel has a rotational inertia of 3.0 kg⋅m2. At the instant the wheel has a counterclockwise angular velocity of 10.0 rad/s, an average counterclockwise torque of 6.0 N⋅m is applied, and continues for 4.0 s. What is the magnitude of the change in angular momentum of the wheel?
I = 3
omega = 10 + alpha * t
torque = I * alpha = 3 * alpha
so
6 = 3 * alpha
alpha = 2 rad/s^2
so
omega = 10 + 2 * 4= 18 rad/s
change in I omega = 3 (18 - 10) = 24 kg m^2/sec
To find the change in angular momentum of the wheel, we need to use the formula:
ΔL = τ * Δt
Where:
ΔL is the change in angular momentum
τ is the torque applied
Δt is the change in time
Given:
Rotational inertia (I) = 3.0 kg⋅m²
Angular velocity (ω) = 10.0 rad/s
Torque (τ) = 6.0 N⋅m
Time (Δt) = 4.0 s
To calculate the change in angular momentum, we first need to find the initial angular momentum (L_initial) and the final angular momentum (L_final).
The formula for angular momentum (L) is:
L = I * ω
For the initial angular momentum (L_initial), substitute the given values into the formula:
L_initial = I * ω
= 3.0 kg⋅m² * 10.0 rad/s
= 30.0 kg⋅m²/s
Next, we can calculate the final angular momentum (L_final) by adding the change in angular momentum (ΔL) to the initial angular momentum:
L_final = L_initial + ΔL
Since we are given the torque (τ) and the change in time (Δt), we can calculate the change in angular momentum (ΔL) using the formula mentioned above:
ΔL = τ * Δt
= 6.0 N⋅m * 4.0 s
= 24.0 N⋅m⋅s
Now we can calculate the final angular momentum (L_final):
L_final = L_initial + ΔL
= 30.0 kg⋅m²/s + 24.0 N⋅m⋅s
= 54.0 N⋅m⋅s
The magnitude of the change in angular momentum of the wheel is 54.0 N⋅m⋅s.
To calculate the change in angular momentum of the wheel, we need to find the initial and final angular momentum and then subtract them.
The formula for angular momentum is L = Iω, where L is angular momentum, I is the rotational inertia, and ω is the angular velocity.
Given:
Rotational inertia (I) = 3.0 kg⋅m^2
Initial angular velocity (ω_initial) = 10.0 rad/s
Average counterclockwise torque (τ) = 6.0 N⋅m
Time (t) = 4.0 s
First, let's find the initial angular momentum:
Initial angular momentum (L_initial) = I ω_initial
Substituting the given values:
L_initial = (3.0 kg⋅m^2) * (10.0 rad/s)
Calculate:
L_initial = 30.0 kg⋅m^2/s
Next, let's find the final angular momentum. Since a constant torque is applied for a fixed amount of time, the change in angular velocity (∆ω) can be calculated using the formula τ = ∆L/∆t, where ∆L is the change in angular momentum and ∆t is the change in time.
Rearranging the formula, we have ∆L = τ * ∆t:
∆L = (6.0 N⋅m) * (4.0 s)
Calculate:
∆L = 24.0 N⋅m⋅s
Now let's find the final angular momentum:
L_final = L_initial + ∆L
Substituting the values:
L_final = 30.0 kg⋅m^2/s + 24.0 N⋅m⋅s
Calculate:
L_final = 54.0 kg⋅m^2/s
Finally, to find the magnitude of the change in angular momentum, we subtract the initial angular momentum from the final angular momentum:
∆L = |L_final - L_initial|
Calculate:
∆L = |54.0 kg⋅m^2/s - 30.0 kg⋅m^2/s|
∆L = 24.0 kg⋅m^2/s
Therefore, the magnitude of the change in angular momentum of the wheel is 24.0 kg⋅m^2/s.