Rotational Mechanics: Calculating Torque as a Function of Time

In summary, a hamster running on an exercise wheel expends a torque on the wheel. The torque is determined by the angular velocity and the moment of inertia.
  • #1
PoofyHair
4
0
Hello,

Hopefully this is in the correct place.
I am presented with a the following problem.
"A hamster running on an exercise wheel, exterts a torque on the wheel. If the wheel has an angular velocity that can be expressed as:
[tex]\omega[/tex](t)= 3.0 rads/s + (8.0 rad/s[tex]^{}2[/tex])t + (1.5 rad/s[tex]^{}4[/tex])t[tex]^{}3[/tex]. Calculate the torque on the wheel as a function of time. Assume that the moment of inertia is 500 kg*m[tex]^{}2[/tex] and is constant."

[tex]\tau[/tex]=Fr F=m[tex]\alpha[/tex] and I=mr[tex]^{}2[/tex]

I then said that [tex]\tau[/tex]=m[tex]\alpha[/tex]r. Next I set I=mr[tex]^{}2[/tex] equal to m and plugged it into [tex]\tau[/tex]=m[tex]\alpha[/tex]r.
I got [tex]\tau[/tex]=I[tex]\alpha[/tex]/r.

After that I differentiated the angular velocity and got [tex]\alpha[/tex](t)=8.0 + 3(1.5)t[tex]^{}2[/tex]. I plugged it in [tex]\tau[/tex]=I[tex]\alpha[/tex]/r and solved. My end result is: [tex]\tau[/tex](t)=2250t[tex]^{}2[/tex] + 4000[tex]/[/tex]r.

Is this correctly done?
 
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  • #2
PoofyHair said:
F=m[tex]\alpha[/tex]
One error is mixing up Newton's 2nd law for rotation and translation.
For translation:
[tex]F = m a[/tex]
For rotation:
[tex]\tau = I \alpha[/tex]
 
  • #3
Ok, thank you very much.
 
  • #4
PoofyHair said:
Hello,

Hopefully this is in the correct place.
I am presented with a the following problem.
"A hamster running on an exercise wheel, exterts a torque on the wheel. If the wheel has an angular velocity that can be expressed as:
[tex]\omega[/tex](t)= 3.0 rads/s + (8.0 rad/s[tex]^{}2[/tex])t + (1.5 rad/s[tex]^{}4[/tex])t[tex]^{}3[/tex]. Calculate the torque on the wheel as a function of time. Assume that the moment of inertia is 500 kg*m[tex]^{}2[/tex] and is constant."

[tex]\tau[/tex]=Fr F=m[tex]\alpha[/tex] and I=mr[tex]^{}2[/tex]

I then said that [tex]\tau[/tex]=m[tex]\alpha[/tex]r. Next I set I=mr[tex]^{}2[/tex] equal to m and plugged it into [tex]\tau[/tex]=m[tex]\alpha[/tex]r.
I got [tex]\tau[/tex]=I[tex]\alpha[/tex]/r.

After that I differentiated the angular velocity and got [tex]\alpha[/tex](t)=8.0 + 3(1.5)t[tex]^{}2[/tex]. I plugged it in [tex]\tau[/tex]=I[tex]\alpha[/tex]/r and solved. My end result is: [tex]\tau[/tex](t)=2250t[tex]^{}2[/tex] + 4000[tex]/[/tex]r.

Is this correctly done?

NO

L=IA=Id(w)/dt=I(8+9/2t^2)
A=angular acceleration
w=angular velocity
I=momentum of inertia=mr^2
 

1. What is rotational mechanics?

Rotational mechanics is the branch of classical mechanics that deals with the motion of objects that rotate around a fixed axis. It involves the study of torque, angular velocity, and angular acceleration.

2. How is torque calculated?

Torque is calculated by multiplying the force applied to an object by the distance from the axis of rotation. The mathematical formula for torque is T = F x r, where T is torque, F is force, and r is the distance from the axis of rotation.

3. What is the relationship between torque and angular acceleration?

Torque and angular acceleration have a direct relationship, meaning an increase in torque will result in an increase in angular acceleration. This relationship is described by the formula T = Iα, where T is torque, I is the moment of inertia, and α is the angular acceleration.

4. How do you calculate torque as a function of time?

To calculate torque as a function of time, you need to know the angular acceleration and moment of inertia of the rotating object. Then, you can use the formula T = Iα to calculate the torque at a specific time. Alternatively, if you have the angular velocity as a function of time, you can use the formula T = I(dω/dt) to calculate the torque at that time.

5. What are some real-life applications of rotational mechanics?

Rotational mechanics has many real-life applications, including the design of engines, turbines, and other rotating machinery. It is also used in sports, such as figure skating and gymnastics, to understand and improve the performance of rotational movements. Rotational mechanics is also essential in understanding the motion of objects in space, such as planets and satellites.

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