What is the angular momentum of a 2.5 kg uniform cylindrical grinding wheel of radius 19 cm when rotating at 1200 rpm? And how much torque is required to stop it in 5.7 s? Need help with step-by-step explanation.
To calculate the angular momentum of a rotating object, you will need to use the formula:
Angular momentum (L) = Moment of inertia (I) x Angular velocity (ω)
Step 1: Calculate the moment of inertia (I)
The moment of inertia for a uniform cylindrical grinding wheel can be found using the formula:
I = 0.5 x m x r^2
where
I = moment of inertia
m = mass of the object
r = radius of the object
Given that the mass of the grinding wheel is 2.5 kg and the radius is 19 cm (or 0.19 m), we can substitute these values into the formula:
I = 0.5 x 2.5 kg x (0.19 m)^2
I = 0.5 x 2.5 kg x 0.0361 m^2
I = 0.04525 kg·m^2
Step 2: Convert the angular velocity to radians per second
The angular velocity is given in revolutions per minute (rpm), but we need it in radians per second (rad/s) in order to use the formula for angular momentum. Since 1 revolution is equal to 2π radians, we can use the conversion factor:
1 rpm = (2π rad) / (60 s)
Given that the angular velocity is 1200 rpm, we can substitute this into the conversion factor:
ω = (1200 rpm) x (2π rad) / (60 s)
ω = 40π rad/s
Step 3: Calculate the angular momentum
Using the formula for angular momentum:
L = I x ω
L = 0.04525 kg·m^2 x 40π rad/s
L ≈ 5.69 kg·m^2/s
So, the angular momentum of the grinding wheel is approximately 5.69 kg·m^2/s.
To calculate the torque required to stop the grinding wheel, you can use the formula:
Torque (τ) = Change in angular momentum (ΔL) / Time (t)
Given that the time to stop the wheel is 5.7 seconds:
Step 4: Calculate the change in angular momentum
Since the wheel is initially rotating and we want to stop it, the change in angular momentum (ΔL) is equal to the negative value of the angular momentum:
ΔL = - L
ΔL = -5.69 kg·m^2/s (from the previous step)
Step 5: Calculate the torque
Using the formula for torque:
τ = ΔL / t
τ = -5.69 kg·m^2/s / 5.7 s
τ = -0.9982 N·m
So, the torque required to stop the grinding wheel in 5.7 seconds is approximately -0.9982 N·m. The negative sign indicates that the torque is applied in the opposite direction of the initial rotation.
To find the angular momentum of the grinding wheel, we can use the formula:
Angular momentum = moment of inertia * angular velocity
1. First, we need to calculate the moment of inertia of the grinding wheel. The moment of inertia for a uniform cylindrical object is given by the formula:
Moment of inertia = (1/2) * mass * radius^2
So, substituting the given values:
Mass (m) = 2.5 kg
Radius (r) = 19 cm = 0.19 m
Moment of inertia = (1/2) * 2.5 kg * (0.19 m)^2
2. Next, we need to calculate the angular velocity in radians per second. Since the given angular velocity is in revolutions per minute (rpm), we need to convert it to radians per second. There are 2π radians in one revolution, and 60 seconds in one minute, so the conversion factor is:
Angular velocity (ω) = (2π/60) * 1200 rpm
3. With the moment of inertia and angular velocity calculated, we can find the angular momentum:
Angular momentum = moment of inertia * angular velocity
Plug in the values:
Angular momentum = (1/2) * 2.5 kg * (0.19 m)^2 * [(2π/60) * 1200 rpm]
4. Solve the equation to find the angular momentum.
Now, to determine the torque required to stop the grinding wheel, we can use the following equation:
Torque = rate of change of angular momentum
Torque = (change in angular momentum) / (time to stop)
1. To find the change in angular momentum, subtract the initial angular momentum from the final angular momentum. Since the wheel is being stopped, the final angular momentum is 0.
Change in angular momentum = 0 - initial angular momentum
2. The time taken to stop is given as 5.7 seconds.
3. Now, substitute the values into the equation:
Torque = (change in angular momentum) / (time to stop)
Solve the equation to find the torque required. This will give you the step-by-step explanation of finding the angular momentum and torque required for the given grinding wheel.
chose your moment of Inertia model. I think solid disk is a good model. I= 1/2 mr^2
angMomentum= I*angvelocity
where w (angular velocity)=2PI*1200/60 rad /seconds
Torque=momentinertia*angularacceleration
= 1/2 m r^2 * change in angvelocity/time
= 1/2 m r^2 * (wf-wi)/5.7sec
wf=0 ; wi is 2PI*1200/60 rad/sec