A wheel rotates without friction about a stationary horizontal axis at the center of the wheel.A constant tangential force of 75N is applied to the rim of the wheel.The wheel has a radius of 0.13m.The wheel has an angular speed of 14.5rev/s after 3.70s.What is the momentum of inertia of the wheel if it's starting from rest?
moment= torqe/angular scceleartion
Torque=75*.13 N-m
angular acceleration= 14.5*2PI/3.7 rad/sec
Why did the wheel go to school to learn about inertia? Because it wanted to become a "momentum of inertia"! Now, let me calculate the momentum of inertia for you.
The formula for the momentum of inertia of a rotating object is given by:
I = (1/2) * m * r^2
Where:
I = momentum of inertia
m = mass of the object
r = radius of the object
In this case, we need to find the momentum of inertia of the wheel. Since the wheel is rotating about a stationary axis, we don't need the mass of the wheel. We can just use the formula I = m * r^2.
Given:
r = 0.13m
Now, to find the angular speed, let's use the formula:
ω = Δθ/Δt
Where:
ω = angular speed (in radians per second)
Δθ = change in angle (in radians)
Δt = change in time (in seconds)
We are given that the wheel has an angular speed of 14.5 rev/s after 3.70s. Let's convert revolutions to radians:
1 revolution = 2π radians
So, the angular speed is:
ω = 14.5 rev/s * 2π rad/rev = 29π rad/s
Now, we can find the momentum of inertia using the formula:
I = m * r^2
Since the wheel is starting from rest, meaning its initial angular speed is 0, there is no initial momentum of inertia. Instead, we need to find the final momentum of inertia. We'll use the formula:
I = (m * r^2) / 2 = (I1 * ω1) / ω
Where:
I1 = initial momentum of inertia (0 in this case)
ω1 = initial angular speed (0 in this case)
ω = final angular speed (29π rad/s in this case)
Now, let's calculate the final momentum of inertia:
I = (0 * 0) / (29π rad/s) = 0
So, the momentum of inertia of the wheel, starting from rest, is 0. I guess it's inertia's way of saying "no momentum, no problem!"
To find the momentum of inertia of the wheel, we can use the equation:
angular momentum = moment of inertia × angular speed
Initially, the wheel is at rest, so the initial angular speed is zero. The final angular speed is given as 14.5 rev/s.
We can convert the rev/s to rad/s by multiplying it by 2π:
final angular speed = 14.5 rev/s × 2π rad/rev = 14.5 × 2π rad/s
The change in angular speed is the final angular speed minus the initial angular speed, so:
change in angular speed = 14.5 × 2π rad/s
The time is given as 3.70 s.
Now we can calculate the change in angular speed per second:
change in angular speed per second = change in angular speed / time = (14.5 × 2π rad/s) / 3.70 s
Next, we can calculate the torque applied to the wheel using the equation:
torque = force × radius
The force is given as 75 N and the radius is 0.13 m:
torque = 75 N × 0.13 m
Now we can use the torque and change in angular speed per second to find the moment of inertia:
moment of inertia = torque / (change in angular speed per second)
Substituting the values, we get:
moment of inertia = (75 N × 0.13 m) / ((14.5 × 2π rad/s) / 3.70 s)
Now we can calculate the moment of inertia using a calculator:
moment of inertia = 3.922 kg·m^2 (rounded to 3 decimal places)
Therefore, the momentum of inertia of the wheel, starting from rest, is 3.922 kg·m^2.
To find the moment of inertia of the wheel when it's starting from rest, we need to use the concepts of rotational motion and torque.
Step 1: Calculate the angular acceleration
Given that the wheel is starting from rest, the initial angular velocity, ω₀, is 0. The final angular velocity after 3.70s, ω, is given as 14.5 rev/s. We can convert this to rad/s by multiplying it by 2π (since 1 revolution is equal to 2π radians).
ω = 14.5 rev/s * 2π rad/rev ≈ 91.0 rad/s
The angular acceleration, α, can be calculated using the equation:
α = (ω - ω₀) / t
α = (91.0 rad/s - 0) / 3.70 s ≈ 24.6 rad/s²
Step 2: Calculate the torque
The torque, τ, acting on the wheel can be calculated using the equation:
τ = I * α
where I is the moment of inertia and α is the angular acceleration. Rearranging the equation, we get:
I = τ / α
Given that a constant tangential force, F, of 75 N is applied at the rim of the wheel and the radius, r, is 0.13 m, we can find the torque using the equation:
τ = F * r
τ = 75 N * 0.13 m = 9.75 N·m
Step 3: Calculate the moment of inertia
Now, substituting the values into the equation, we get:
I = τ / α
I = 9.75 N·m / 24.6 rad/s²
I ≈ 0.397 kg·m²
So, the moment of inertia of the wheel when starting from rest is approximately 0.397 kg·m².