The length of the moment arm of a force is the distance from the point where the force is applied to the point where the resulting rotation or movement occurs.
In simple terms, it is how far you have to pull in order to make something happen. For example, if you have to pull one meter to open a door, then the length of the moment arm of the force used to open the door is one meter.
Calculating the length of the moment arm of a force is not an easy task, which is why it is important to know how to identify it. Knowing its size can help you understand and assess how much effect a specific force has on an object.
This article will discuss more about what the length of the moment arm of the force is and how to identify it.
Calculate the length of the moment arm r
The length of the moment arm can be calculated by finding the length of the muscle, ligament, or tendon, and then adding the length of the joint it crosses.
If the bicep curls a weight up to your shoulder, then the muscle is the length of the bicep, and the joint crossed is the shoulder. The total moment arm would be the length of the bicep plus the length of the shoulder.
The more distal (away from the center point) a muscle or tendon is, then the greater effect it will have on a joint rotation. This is because it takes longer for torque transmitted down it to reach the joint.
If a muscle or tendon is shorter but located more distally, it will have a greater effect than a longer one that is located more proximally.
Find the point P
The first step is to find the point P. This can be done in a few ways, depending on what type of problem you are solving.
If you are solving this problem for a physics question, then you need to find the axis of rotation and the point of action. The axis of rotation is easily found, it is the line that the weight hangs from. The point of action is where the force acts, or where the weight drops.
If you are solving this problem for art, then you need to find where you want the drop to hang from and where you want it to drop. The rest can be determined with art tricks!
To solve this problem for both physics and art, we will first determine where the drop hangs from and then where it drops.
Find the direction of F⃗
The first step is to find the direction of the applied force. You can do this by drawing a line from point A to point B, where A is the shoulder and B is the wrist.
If the force is coming from behind your shoulder, then it would be directed toward your back. If it’s coming from in front of your shoulder, then it would be directed toward your front.
If it’s coming from above your shoulder, then it would be directed downward. If it’s coming from below your shoulder, then it would be directed upward.
The next step is to find point P, which in this case is the bony prominence at the base of the thumb. Next, draw a line from point P to point A on the shoulder joint axis.
Calculate angular displacement θ
The next step is to calculate the angular displacement, or how far the limb moves in rotation. To do this, you need to know the length of the muscle-tendon unit.
Muscle-tendon units have what’s called a physiological length, which is where they can exert the most force and elongate the most. When a muscle contracts, it elongates; when it relaxes, it shortens.
Physiological length depends on many factors: how long the muscle fibers are, how thick the connective tissue surrounding them is, whether the muscle is relaxed or contracted, and so on.
Because physiological length changes depending on these factors, it’s important to measure it when assessing explosive performance—and not just for muscles and tendons. Other structures that can have an effect on explosivity include gravity platforms (GP) and jumping bags (JB).
Calculate linear displacement r
The next step is to calculate the linear displacement, or length, of the mass that was displaced.
You already calculated the radius of the circle in the previous section, so all you have to do is solve for r, the displacement.
Linear displacement, r, is found by taking the square root of mass times velocity. You already calculated mass and velocity in previous sections, so all you have to do is combine these values and you have your answer!
Remember: Always check your work! It is easy to make a small mistake when working with equations, so always make sure that you are using the correct values.
This section can be tricky because it requires understanding of square roots. If you need more help, ask a friend or family member to help you go over this section with you.
What is length, rm, of the moment arm of force F⃗ about point P?
When force is applied to a lever arm, or moment arm, the amount of force that is applied to the object changes based on how far away from the point of application the force is.
A longer arm will have a higher amount of force applied to it due to a longer exposure to the applying point. A shorter arm will have a lower amount of force applied to it due to a shorter exposure to the applying point.
For example, if you were trying to pick up a heavy box and could not get any closer to it than one foot, you would need enough strength to hold up the box for only one foot before it fell. If you were able to get closer than one foot, then you would be able to hold up the box for more than one foot before it fell.
The length of the moment arm depends on two things: where the force is applied and where it is not applied.
What is angle θ?
The third component of the torque is called the angle. This is the degree of rotation that occurs as a result of the force.
If the force is directed perpendicular to the point of rotation, then there is no angle and this component of torque is zero. If there is a small angle, then this component increases, thus causing more rotation.
Imagine trying to pick up a box on the floor using both hands at a slight angle to each other. You would have to use more force to get the box moving than if you had straight arms. This is because you are using more of an angle in your arms when grasping the box, which causes more resistance.
Similar things happen in sports when trying to stop or turn something or someone else around. The greater the degree of rotation (angle) you can generate, the better.
What is point P?
Point P is the point about which the force acts. In our case, point P is the initial position of the muscle and where the external force acts.
In our example, point P is where the bicep contracts and pulls the forearm. The length of the moment arm of a force about point P depends on where point P is relative to where the muscle contracts and where the force acts.
If point P is very close to where the muscle shortens, then the length of the moment arm of the force F⃗ will be small. This is because there is not much distance between point P and where the muscle shortens.
If point P is far from where the muscle shortens, then there will be a longer length of the moment arm of force F⃗. This is because there would be more distance between point P and wherethe muscle shortens.
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