Suppose you start an antique car by exerting a force of 300N on its crank for .250 s. What angular momentum (in kgm^2/s ) is given to the engine if the handle of the crank is .300m from the pivot and the force is exerted to create maximum torque the entire time?
The angular momentum (L) of an object is given by the product of its moment of inertia (I) and its angular velocity (ω). In this case, we need to calculate the angular momentum given to the antique car's engine.
The torque (τ) exerted on the engine is the product of the force applied (F) and the perpendicular distance from the pivot point to the line of action of the force (r). In this case, the force applied is 300 N, and the distance from the pivot point to the handle of the crank is 0.300 m. Assuming maximum torque is created, we can use τ = Fr to determine the torque.
The torque exerted on the engine can also be expressed as τ = Iα, where α is the angular acceleration. Since the force is exerted for a short amount of time (.250 s), we can consider the applied force as resulting in an impulse, which causes an angular acceleration α.
The equation for the torque can be rearranged as I = τ/α. We can use this equation to find the moment of inertia of the engine.
Now, let's calculate step by step:
1. Calculate the torque exerted:
τ = Fr = 300 N * 0.300 m = 90 Nm
2. Find the moment of inertia (I):
I = τ/α
To calculate α, we can use the impulse-momentum relationship:
Impulse (J) = F * Δt = m * Δv
Here, Δt = 0.250 s
The car's mass (m) and the change in angular velocity (Δω) can be determined using the relationship:
J = I * Δω
We can rearrange the equation as:
Δω = J/I = F * Δt/I
Using the given values,
Δω = (300 N * 0.250 s) / I
Now, equating the equations for Δω:
(300 N * 0.250 s) / I = Δω = (2π radians/s)
Simplifying this equation, we get:
I = (300 N * 0.250 s) / (2π radians/s) = 119.79 kgm^2
3. Calculate the angular momentum (L):
L = I * ω
Since the maximum torque is created the entire time, the angular velocity (ω) is constant.
Therefore, the angular momentum given to the engine is 119.79 kgm^2/s.