Angular Velocity and Power Input Question

[itsgiiinge]

Hi there,

I've got these questions to do, and I've looked at the formulas and can't seem to plug in the numbers with the values I've been given. I think I'm missing something somewhere, if someone could just start me off, or be so kind as to go through the method so I know for the future?

Thank you :)

a) A wheel rolls up an incline of 15 degrees on its hubs without slipping and is pulled by the 100N force applied to the cord wrapped around its outer rim. The wheel starts from rest, so compute its angular velocity after its center has moved a distance of 3m up the incline. Radius to inner rim is 100mm, radius to outer rim 200mm.

b) The wheel has a mass of 40 kg with center of mass at O and has a centroidal radius of gyration of 150mm. Determine the power input from the 100N force at the end of the 3m motion interval.

 

[Simon Bridge]

Hi there,

I've got these questions to do, and I've looked at the formulas and can't seem to plug in the numbers with the values I've been given. I think I'm missing something somewhere,

Yep - plugging numbers into equations does not work ... you are missing the physics - get that right and the equations will take care of themselves.

if someone could just start me off, or be so kind as to go through the method so I know for the future?

You should still show us how you are thinking about this so we can target the specific place you need help.

a) A wheel rolls up an incline of 15 degrees on its hubs without slipping and is pulled by the 100N force applied to the cord wrapped around its outer rim. The wheel starts from rest, so compute its angular velocity after its center has moved a distance of 3m up the incline. Radius to inner rim is 100mm, radius to outer rim 200mm.

Hint: It's rolling without slipping - so it's angular velocity and speed up the ramp are related. The force is applied at the rim - so it will make a torque.

b) The wheel has a mass of 40 kg with center of mass at O and has a centroidal radius of gyration of 150mm. Determine the power input from the 100N force at the end of the 3m motion interval.

Power is energy over time... the energy comes from the work done by the force.

 

[itsgiiinge]

Hi, thanks for your response.

Torque. i knew there was something I was missing.

So kind of working backwards, to work out torque I have the formula T = rFsinθ. Would the radius be the 200mm to the outside rim from the centre? and what is the centroidal radius of gyration because I've never heard the term before.

So once I have the torque, the power would then be P = T x ω

But to find the angular velocity ω, the formula I have is ω = v/r which is what I was given. Is v some sort of linear velocity?

I'm very sorry, i know this is simple but I haven't done these sorts of problems since I was in school.

Thanks again for your help

EDIT: I'm using the formulas I was given, but if there are better ones, that'd be great.

 

[Simon Bridge]

"Centroid of gyration" is related to "moment of inertia" ... look it up ;)

Don't use formulas!
You need to be able to understand the physics of what is going on - a torque is a twisting force. If it is created by a linear force $\vec{F}$ acting at some position $\vec{r}$ from the center of rotation, then the torque is the cross product of these: $\vec{\tau}=\vec{r} \times \vec{F}$ . So torque is a (pseudo-)vector.

$\omega$ is the angular speed - under an unbalanced torque, this will change. So your equation for power isn't going to work right?

Angle is the rotational version of position, rate of change of angle is $\omega = d\theta /dt$ is speed, $\alpha = d\omega /dt$ is acceleration and force is $\tau = I \alpha$ so I plays the role of inertia (mass) and it is called "moment of inertia" for historical reasons.

I don't actually remember these equations - I just reconstructed them right now from the physics. All I did was describe the ideas using math like a language.

I think you need to brush up on your physical understanding:
http://www.sparknotes.com/physics/rotationalmotion/rotationaldynamics/section3.rhtml
http://www.phy.ntnu.edu.tw/ntnujava/index.php?topic=1025.0
http://physics.info/rolling/summary.shtml

 

[rezeew]

Hi there,

Sure, I'd be happy to help you with these questions. Let's start with the first one.

a) To solve this question, we can use the formula for angular velocity, which is ω = v/r, where ω is the angular velocity, v is the linear velocity, and r is the radius. We can also use the formula for linear velocity, which is v = ωr, where ω is the angular velocity and r is the radius.

First, let's find the linear velocity of the wheel. Since the wheel is rolling without slipping, we can use the formula v = ωr, where ω is the angular velocity and r is the radius. We are given the radius to the inner rim, which is 100mm, so we need to convert it to meters by dividing it by 1000. This gives us a radius of 0.1m. Now, we can find the linear velocity by plugging in the values into the formula: v = ωr = ω(0.1) = 0.1ω.

Next, we need to find the angular velocity. We can use the formula ω = v/r, where ω is the angular velocity, v is the linear velocity, and r is the radius. We are given the linear velocity, which we just calculated to be 0.1ω, and we are also given the radius to the outer rim, which is 200mm. Again, we need to convert this to meters by dividing it by 1000, giving us a radius of 0.2m. Now, we can plug in the values into the formula: ω = v/r = (0.1ω)/0.2 = 0.5ω.

We now have two equations for the angular velocity: ω = 0.1ω and ω = 0.5ω. We can solve for ω by setting these two equations equal to each other: 0.1ω = 0.5ω. Solving for ω, we get ω = 0.5 rad/s.

b) To solve this question, we can use the formula for power, which is P = Fv, where P is the power, F is the force, and v is the velocity.

First, let's find the velocity of the wheel. We know that the wheel has moved a distance