Mechanical Engineering

  

 

 

 

Mechanics : Torque

 

Torque is a measure of how much ration can be made by the force (linear force) acting upon a specified point.  Many people have difficulties in clearly understanding the concept of torque. In some case, they do not have problems in calculating the torgue in give situation but still have difficulties in understanding the real meaning of the calculation.

So, before getting into the mathematical definition or calculation, I would try to give you some intuitive understanding on what Torque is.

 

I think one of the biggest reason for the trouble understanding the torque is because they are confused with Torque and Force. Also, they tend to think that if you apply larger force and you would see larger torque.

 

In short, Torque and Force are related but they are not the same, and large force applied to an object does not necessarily produce larger torque.

 

Let's assume that you have a wrench plugged around a bolt and applied the same amount of force in various different directions as shown below and ask yourself a question 'Will this force rotate the bold ?' for each of the case.

 

How about case (A) ?  Will the applied force on the wrench rotate the bolt ? You know the answer is 'No'. However strong force you applied in this direction, the bold would not rotate. It means that the applied force in this case does not generate any rotating force. In other words, we say the Torque at the point of the bolt is zero.

How about case (B), (C) ? Will the applied force on the wrench rotate the bolt ? For the intuition (or experience), you would know that these force would generate the rotation. Even if the bold may not rotate(turn), at least some rotational force will be generated. In this case, we say Torque will be generated by the force F.

 

 

Now let's think of this rotational force (Torque) in a little bit different aspect. Let's suppose we apply the same amount of the force in different angles as shown below. Which case would generate the strongest rotational force(Torque) ? From your experience (or intuition), you would know that (B) would generate the stronger Torque than (A), and (C) would generate stronger torque than (B). It means that the amount of torque is related to the angle between the arm and the force applied to the arm.

 

 

 

Torque is a vector

 

Even though you may learned as if Torque is a kind of Scalar in high school physics, the Torque is a vector in reality. It means that we need to think of not only the amount (size) of the Torque but also the direction of it.

To clarify on this, let's express the situation in more formal way as sown below. I represented the arm as a vector r and the force applied to the arm as a vector F. In this case, the position and direction of the torque generated by r and F can be represented as t (tau) as in (B).  As you see, the starting point of the torque vector is located at the pivot point of the rotation and the direction of the torque vector is perpendicular of the plane formed by the two vector r and F.

Now you may ask...  we can have two different directions of the vector that is perpendicular to the plane formed by the two vector r and F, Upwards vector and downwards vector in this case. Which one is the correct one ? The correct direction of the torque vector is determined by the right hand rule as shown in (B) and (C).

 

 

 

How to calculate the Torque ?

 

In high school physics, you may have learned the torque equation as follows. As you would notice, this is a scalar equation. Very strictly speaking, this is not the accurate formula because Torque is not a scalar, but a vector. However, we used this equation in high school physics with a few assumption (even though the assumption may not be explicitely mentioned in the text book). The assumptions are

  • The direction of the force in this equation is perpendicular to the direction of the arm
  • The direction of the Torque vector is the direction determined by the right hand rule

 

 

 

In college level physics or engineering course, you would see the torque equation in vector form as shown below. As you see here, all the components in the equation are vectors and operator is 'Cross Product', not the scalar multiplicatioin. With this, all the assumptions mentioned above (used in high school physics) are explictely expressed in the definiton of Cross Product (I strongly recommend you to read through Cross Product page if you are not familiar with this operation)

 

 

 

 

Example 01 >

 

Let's suppose a situation where you want to unscrew a car tire using a wrench as shown below. In order make jobs easier, you would decided to use foot in stead of your arm/hands. Let's assume that you are pressing down the end of the wrench arm with 200 N. What will the torque generated by the wrench at the point of the bolt ?

 

 

 

Step 1 > Define your reference coordinate system and direction

 

When you try to solve any kind of mechnical or dynamic system, the first thing you have to do is to define a reference coordinate system. This is important because the sign and value of all the elements for the vectors that will be used in the problem solving process is determined by this reference coordinate system. It is completely up to you on how to define the reference as long as you will apply it consistantly throughout the whole problem solving process.

I set the reference coordinate system as shown below. The axis and arrows reference the positive direction of each translational and rotational vectors.

 

 

 

Step 2 > Visualize the vectors

 

Draw all the vectors given to you. In the following picture, the red vector indicates the distance vector (It is the vector generated by the arm of the wrench). The blue vector indicates the force applied to the end of the wrench arm.

 

 

 

Step 3 > Define the vectors with real number or mathematical symbols given to you

 

The two vectors F and r can be defined in numbers given to us in the problem. They can be defined as follows.

 

 

 

Step 3 > Do the math now.

 

Now the remaining thing is just doing the math as follows. To do this, you should have the knowledge of two important mathematical concept and operators, Cross Product and Determiant. If you are not familiar with these concept, refer to Cross Product page and Determinant page.

 

 

 

Step 4 > Visualize the result and check if it make sense

 

Many people think just calculating the math equation is enough for solving the problem. However, in some case you may end up with some solutions that would make sense mathematically but does not make sense in reality. Or you really came up with correct numbers both in mathematics and in reality, but you don't understand the meaning of those numbers. So I recommend you to interpret those numbers in the practical situations given to you. In case of this kind of vector calculation problem, draw the resulting vectors and numbers onto the problem scene given to you and check if those numbers and vectors make sense.

 

 

 

 

Reference :

 

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