What torque applied to a disk with radius 7.0cm for 3.5s will produce an angular momentum of 25Nms?
To calculate the torque applied to a disk, we can use the equation for angular momentum:
Angular momentum (L) = moment of inertia (I) × angular velocity (ω)
The moment of inertia for a disk is given by:
Moment of inertia (I) = (1/2) × mass (m) × radius squared (r^2)
Given that the radius of the disk is 7.0 cm (0.07 m), and the angular momentum required is 25 Nms, we can rearrange the equations to solve for torque.
Angular momentum (L) = (1/2) × mass (m) × radius squared (r^2) × angular velocity (ω)
The angular velocity can be calculated using the formula:
Angular velocity (ω) = Change in angular displacement / Change in time
As there is no information given about the angular displacement, we assume it to be a full revolution (2π radians) in the time period given (3.5 seconds). Therefore, the angular velocity is:
Angular velocity (ω) = (2π radians) / (3.5 seconds)
Now, substituting the given values into the equation for angular momentum:
25 Nms = (1/2) × m × (0.07 m)^2 × [(2π radians) / (3.5 seconds)]
Simplifying the equation, we can solve for mass:
m = [(25 Nms) / [(1/2) × (0.07 m)^2 × (2π radians) / (3.5 seconds)]]
m ≈ 4.943 kg
Now, we can calculate the moment of inertia of the disk:
Moment of inertia (I) = (1/2) × m × (0.07 m)^2
Substituting the calculated mass:
I = (1/2) × 4.943 kg × (0.07 m)^2
I ≈ 0.012 kg·m²
Finally, we can calculate the torque applied to the disk using the formula:
Torque (τ) = Moment of inertia (I) × angular acceleration (α)
However, as no information about the angular acceleration is given, we assume it to be constant as the disk reaches the required angular velocity during the 3.5 second period. Therefore, the angular acceleration is:
Angular acceleration (α) = Angular velocity (ω) / Time (t)
Substituting the known values:
α = (2π radians) / (3.5 seconds)
Now, substituting into the equation for torque:
τ = (1/2) × 0.012 kg·m² × [(2π radians) / (3.5 seconds)]
τ ≈ 0.034 N·m
Therefore, a torque of approximately 0.034 N·m is required to produce an angular momentum of 25 N·s in a disk with a radius of 7.0 cm for 3.5 seconds.