# Torque and Moment of Inertia of a Lever Arm

• MattGeo
In summary, the conversation discusses the relationship between torque, angular acceleration, and the moment of inertia in different scenarios such as using a long wrench, opening a door, and swinging a rod. It is noted that as the lever arm or force multiplier increases in size, higher torques can be achieved but at lower angular accelerations. This is also applicable to gears. The additional weight of the tool may also affect the applied torque.
MattGeo
I never really considered this back when I was taking physics in college but imagine for the sake of thought experiment that you have an extremely and impractically long wrench and it is fixed to the bolt you wish to tighten. Now the longer the lever arm the greater the torque so if you double the length of the lever arm you only have to apply half as much tangential force to achieve the same torque, but, as the wrench lever arm becomes more massive and longer, won't the moment of inertia of it begin to impede the tightening of the bolt? Or will the bolt still feel a larger torque but the angular acceleration of the lever arm will be less and less?

Like say when you open a door, you do so by pushing furthest from the hinge because that maximizes the torque, so naturally the doorknob is opposite side of the hinge. Now say the door was twice as wide. It's a uniform door so also say it now has twice the mass. Because the moment of inertia depends on both mass and the square of the radius the moment of inertia will be 8 times as large, but you have doubled the radial distance at which you apply torque so you have 2 times as much torque. So the door now rotates at 1/4 the angular acceleration of the original door? and twice the original torque is still felt at the hinge? It seems like if the lever arm or force multiplier cannot be considered to have negligible size then higher torques are achieved but at lower angular accelerations. Is this true or am I just confused?

Also consider a long rod which is fixed to a hinge mounted to a wall at one end. Sort of like a seesaw where the fulcrum is all the way to edge of one of the ends. The rod starts out horizontal and is allowed to swing down, sweeping out a quarter circle as it becomes vertical. If you make this rod increasingly longer and its moment of inertia increases, will its angular acceleration begin to decrease? It seems to imply that the moment you let go it would take longer to speed up rotationally under the force of gravity. Something about this seems wrong to me but intuition is often wrong.

Any clarification to this would be much appreciated.

Delta2
The additional required torque would be proportional to the desired acceleration of the wrench or door.

##\tau=I\alpha##

In practical situations, the additional weight of the tool, if acting on a close to vertical plane, may have a more noticeable effect on the applied torque.

I sometimes have to open extremely heavy doors with vertical hinge line.
The more noticeable adaptation, compared to opening regular doors, is the reduced angular velocity achieved with increased pulling force.

sophiecentaur
MattGeo said:
It seems like if the lever arm or force multiplier cannot be considered to have negligible size then higher torques are achieved but at lower angular accelerations.
Yes, sure. Same goes for gears, which should not be too big.

## 1. What is torque?

Torque is a measure of the force that causes an object to rotate around an axis. It is calculated by multiplying the force applied to an object by the distance from the axis of rotation.

## 2. How is torque related to the moment of inertia of a lever arm?

The moment of inertia of a lever arm is a measure of an object's resistance to changes in its rotational motion. It is directly proportional to the torque applied to the object, meaning that the greater the moment of inertia, the more torque is needed to cause the object to rotate.

## 3. What factors affect the moment of inertia of a lever arm?

The moment of inertia of a lever arm is affected by the mass and distribution of the object's mass, as well as the distance between the object's axis of rotation and its center of mass.

## 4. How can the moment of inertia of a lever arm be calculated?

The moment of inertia of a lever arm can be calculated by using the formula I = mr^2, where I is the moment of inertia, m is the mass of the object, and r is the distance between the axis of rotation and the object's center of mass.

## 5. What is the significance of the moment of inertia in everyday life?

The moment of inertia is important in everyday life because it helps us understand and predict how objects will behave when a torque is applied to them. It is also essential in the design and engineering of various structures and machines, such as bridges and vehicles.

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