Finding an expression for torque

[Lucy Yeats]

Homework Statement

http://www.physics.ox.ac.uk/users/yassin/mechanics/problems/probs6_2011.pdf

Q 2b.

Homework Equations

The Attempt at a Solution

T is torque and J is angular momentum.
T=dJ/dt=I dω/dt

I tried solving dω/dt=ωk to get dω/dt=ce^kt, but this may not help.

 

[Curious3141]

Homework Statement

http://www.physics.ox.ac.uk/users/yassin/mechanics/problems/probs6_2011.pdf

Q 2b.

Homework Equations

The Attempt at a Solution

T is torque and J is angular momentum.
T=dJ/dt=I dω/dt

I tried solving dω/dt=ωk to get dω/dt=ce^kt, but this may not help.

They represented torque as N.

You're given N = kω ---equation 1

You also know that N = -I$\frac{dω}{dt}$ ---equation 2
(the negative sign is to keep N positive since they're concerned with the numerical value for torque).

You can equate equations 1 and 2 to get kω = -I$\frac{dω}{dt}$

and solve the differential equation. Remember to apply the given bounds using definite integrals on both sides after separating the variables ω and t.

After that, express k in terms of the other variables, and put this back into equation 1 to get N.

 

[Lucy Yeats]

Great, I've got the right answer now.

Thank you very much! :-)

 

[Curious3141]

Great, I've got the right answer now.

Thank you very much! :-)

You're welcome.

 

[asteriatic]

Torque is a measure of the rotational force applied to an object and is defined as the product of the force applied and the distance from the axis of rotation to the point of application of the force. Mathematically, torque can be expressed as T = F × r, where T is torque, F is the force applied, and r is the distance from the axis of rotation to the point of application of the force. This expression shows that the magnitude of torque increases with the force applied and the distance from the axis of rotation. Additionally, torque also has a direction, which is perpendicular to both the force and the distance from the axis of rotation. This can be represented by the right-hand rule, where the direction of the fingers represents the direction of the force and the direction of the thumb represents the direction of the torque.

In terms of angular momentum, torque can be expressed as the time rate of change of angular momentum, which is given by T = dJ/dt = I dω/dt, where J is angular momentum, I is the moment of inertia, and ω is the angular velocity. This equation shows that torque is directly proportional to the moment of inertia and the angular acceleration. It also highlights the relationship between torque and angular momentum, as any change in angular momentum must be accompanied by a corresponding torque.

In summary, torque can be expressed as T = F × r or T = dJ/dt = I dω/dt, depending on the context and the quantities involved. Understanding and being able to calculate torque is essential in the study of mechanics, as it plays a crucial role in rotational motion and the stability of objects.