Matthew has a Master of Arts degree in Physics Education. He has taught high school chemistry and physics for 14 years.
When an object is free to rotate about a fixed axis, a force applied can cause the object to rotate. This lesson deals with the principle of moments, which is the turning effect on an object provided by a force.
The Principle of Moments
The result of a force applied to an object that can rotate is called a moment, which is very similar to torque. The differences are subtle, and the words moment and torque are often interchanged. At any moment during a typical day, whether you realize it or not, you're providing moments quite frequently. Sounds like a riddle doesn't it? How did Pierre Varignon provide moments to his writing utensil in the late 1600s when he wrote down the theory of the principle of moments called Varignon's theorem? That was the last riddle, I promise. Take a moment to clear your mind so we can learn about the principle of moments.
The principle of moments , or Varignon's theorem, states that the net moment about one axis on an object is equal to the sum of the individual moments acting along that axis. In cases where multiple forces are acting, and there is no rotation, the principle of moments is zero.
Enough with the tongue twisting, mind wracking definitions!
The hand is providing a moment on the wrench
We can see that the hand is pulling down on the wrench, and the angle between the force (red arrow) and the wrench is 90o. The angle between the wrench and the force is important because the equation for the moment of force, which is the cross-product or vector product, involves this angle. Moments of force are vectors, which includes a magnitude and a direction.
Equation 1: The bottom equation is for the magnitude of the moment
This is where:
Moment of force has the units newton-meters (Nm)
r is the distance (in meters) from the pivot point to where the force is applied
F is the force applied in newtons (N)
sin is a trigonometric function, and is the ratio of the opposite side of a right triangle to the hypotenuse of the triangle. The sin function is a minimum at 0o, and a maximum at 90o.
θ is the angle between the force and the lever arm in degrees
The direction of the moment can be determined using the right-hand rule. Take your right hand, and make your best '' stick 'em up' '' gun shape. Now, extend your middle finger so it's perpendicular to your pointer finger.
Right-hand rule hand position
Pointer finger points away from pivot point along lever arm
Middle finger points in the direction of the perpendicular force on the lever arm
Thumb points in direction of moment vector
If we apply the right-hand rule to our wrench scenario, the thumb points into the screen. If we had the magnitude of the force, and the distance from the pivot point to where the force is applied, we could get the magnitude of the moment. Let's do two examples with numbers.
A person applies 10 newtons of force 0.5 meter away from the pivot point at 50o on a bank vault lever. What is the moment provided by this force?
Solution: We apply Equation 1 to get the magnitude of the moment, in which we get:
The principle of moments is what we did in one step to get the magnitude of the moment. Let's break it down based on the definition of the principle of moments. The first step is to resolve the force into its components.
Green arrow is parallel force, blue arrow is perpendicular force
The blue arrow can be considered to be acting perpendicularly at the point where the red arrow force is applied. Calculating the magnitude of the perpendicular force we get:
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The green, horizontal force is useless in terms of rotating the lever because Equation 1 tells us to use the sine of the angle between the force and the lever arm, which, in this case, is 0o, and the sine of 0o is 0. Therefore, the moment provided by the parallel-component of the force is 0.
Let's get the perpendicular-component's moment, which is:
The principle of moments states that we add all of the moment components together to get the net moment. Since we know the horizontal moment is 0, we can get the net moment of force, which is also, ultimately:
You can see we get the same answer using both methods. The right-hand rule tells us that the direction of the moment is pointing into the screen, which we will call the negative direction.
Let's pretend that the bank manager realized you didn't turn off the alarm to the bank vault, so he rushes over, and pulls up on the lever with 32 newtons of force angled at 30o 0.25 m away from the pivot point. Will that be enough to stop the lever from rotating, and thus prevent the alarm from going off?
Let's calculate the moment provided by the new force on the lever. As we can see, after plugging in the values, we get:
The moment's vector points out of the screen, and since this is in the opposite direction of the other moment's vector direction we will make it positive. Therefore, the net moment is actually:
Thankfully, the lever won't turn since there is no net moment. Now you won't have to deal with that awful alarm going off, and the police showing up thinking you are robbing the vault!
Let's take a couple of moments to review what we've learned in this lesson about the principle of moments, both its definition and how we calculate it. The result of a force applied to an object that can rotate is called a moment, which is very similar to torque. The principle of moments , or Varignon's theorem, states that the net moment about one axis on an object is equal to the sum of the individual moments acting along that axis. In cases where multiple forces are acting, and there is no rotation, the principle of moments is zero.
The magnitude for a moment is:
In addition, the direction the thumb points using the right-hand rule tells you which direction the moment's vector is pointing.
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