Monday, May 20, 2019

Chapter 7.20 - Principle of Moments

In the previous sectionwe saw conditions of equilibrium of rigid bodies. In this section we will see Principle of Moments

1. In the fig.7.97(a) below, a light (ie. of negligible mass) rod AB lies on the xy-plane
• It’s length is (d1+d2)
Fig.7.97
• It is supported at point 'C' by a 'knife edge support'
• The 'knife edge support' is shown clearly in the 3D view in fig.7.98 below:
Fig.7.98
2. Two forces are applied on the rod
• Those two forces have the following five peculiarities:
(i) The forces do not have the same magnitude
(ii) Both forces have the same directions
    ♦ Let us assume the usual orientation: x-axis is horizontal and y-axis is vertical
    ♦ Then both forces act towards the -ve side of the y-axis
(iii) One force is applied at A
(iv) The other force is applied at B
(v) Both forces are perpendicular to AB
3. As a result of the forces $\mathbf\small{\vec{F}_A}$ and $\mathbf\small{\vec{F}_B}$, a reaction force will develop at C
• This reaction force is denoted as $\mathbf\small{\vec{R}_C}$
• $\mathbf\small{\vec{R}_C}$ is shown to be vertical and directed towards the +ve side of the y-axis
• We will soon prove mathematically that, it is indeed vertical and directed towards the +ve side of y-axis
4. We want the rod AB to remain in equilibrium (both translational and rotational)
• How can we achieve that?
Ans: By ensuring that, the conditions of equilibrium are satisfied
• So let us apply the conditions:
• We have already seen that, if all the forces lie in the xy-plane, we need to check 3 conditions only:
$\mathbf\small{\sum{\vec{F}_{x,ext}}=\vec{0}}$
$\mathbf\small{\sum{\vec{F}_{y,ext}}=\vec{0}}$
$\mathbf\small{\sum{\vec{\tau}_{z,ext}}=\vec{0}}$
• Further, there are no forces in the x-direction. So we need to check 2 conditions only:
$\mathbf\small{\sum{\vec{F}_{y,ext}}=\vec{0}}$
$\mathbf\small{\sum{\vec{\tau}_{z,ext}}=\vec{0}}$
5. First we will apply:
$\mathbf\small{\sum{\vec{F}_{y,ext}}=\vec{0}}$
• If this condition is to be satisfied, the vector sum ($\mathbf\small{\vec{F}_A+\vec{F}_B+\vec{R}_C}$) must be a zero vector
• This condition tells us that, if there is to be translational equilibrium, $\mathbf\small{\vec{R}_C}$ is indeed vertical
• The reason can be written in 5 steps:
(i) If $\mathbf\small{\vec{R}_C}$ is inclined, it will have a vertical as well as a horizontal component
(ii) But we see no other horizontal components to counter it
(iii) So, if there is to be translational equilibrium, $\mathbf\small{\vec{R}_C}$ must be truly vertical
(iv) Also, it must be directed towards the +ve side of the y-axis
(v) Only then we will get ($\mathbf\small{\vec{F}_A+\vec{F}_B+\vec{R}_C}$) equal to a zero vector
6. Since we know the directions, we can write the above equation in terms of unit vectors also:
$\mathbf\small{-|\vec{F}_A|\hat{j}-|\vec{F}_B|\hat{j}+|\vec{R}_C|\hat{j}=\vec{0}}$
$\mathbf\small{\Rightarrow \left(-|\vec{F}_A|-|\vec{F}_B|+|\vec{R}_C|\right)\hat{j}=\vec{0}}$
• In the above equation, we have vectors on both sides. So their magnitudes must be equal
• Thus the condition becomes:
$\mathbf\small{-|\vec{F}_A|-|\vec{F}_B|+|\vec{R}_C|=0}$

7. For applying the next condition, we have to first find the magnitudes and directions of the torques created by the forces in the xy-plane
• Let ‘C’ be the origin. Then we can mark the position vectors as shown in fig.b
• Torque created by $\mathbf\small{\vec{F}_A}$ about C 
$\mathbf\small{(\vec{r}_A \times \vec{F}_A)}$
• Magnitude of this torque
= $\mathbf\small{|\vec{r}_A|\times |\vec{F}_A|\times \sin \theta}$
$\mathbf\small{[d_1\times |\vec{F}_A|\times \sin 90]=d_1\times |\vec{F}_A|}$
(θ = 90o because, forces are perpendicular to the rod)
8. For finding the direction of this torque:
(i) Assume that $\mathbf\small{\vec{F}_A}$ is shifted so that, it's tail end coincides with the tail end of $\mathbf\small{\vec{r}_A}$  
(ii) Imagine that a right handed screw is placed perpendicular to the computer screen
(iii) Turn it from $\mathbf\small{\vec{r}_A}$ to $\mathbf\small{\vec{F}_A}$
(iv) We see that, the screw moves away from the screen, towards us
(v) The rod is lying on the xy-plane. So the computer screen is the xy-plane
(vi) The z-axis is perpendicular to the computer screen
(vii) Let us assume that, the positive direction of the z-axis points towards us
(viii) So we can write:
The torque created by $\mathbf\small{\vec{F}_A}$ is perpendicular to the xy-plane and is directed towards the +ve side of the z axis
• Combining the magnitude and the direction, we can write:
The torque created by $\mathbf\small{\vec{F}_A}$ is: $\mathbf\small{[d_1\times |\vec{F}_A|]\; \hat{k}}$ 
9. Next we find the torque created by $\mathbf\small{\vec{F}_B}$ about C 
$\mathbf\small{(\vec{r}_B \times \vec{F}_B)}$
• Magnitude of this torque
= $\mathbf\small{|\vec{r}_B|\times |\vec{F}_B|\times \sin \theta}$
$\mathbf\small{[d_2\times |\vec{F}_B|\times \sin 90]=d_2\times |\vec{F}_B|}$
(θ = 90o because, forces are perpendicular to the rod)
10. For finding the direction of this torque:
(i) Assume that $\mathbf\small{\vec{F}_B}$ is shifted so that, it's tail end coincides with the tail end of $\mathbf\small{\vec{r}_B}$  
(ii) Imagine that a right handed screw is placed perpendicular to the computer screen
(iii) Turn it from $\mathbf\small{\vec{r}_B}$ to $\mathbf\small{\vec{F}_B}$
(iv) We see that, the screw moves into the screen, away from us
(v) The rod is lying on the xy-plane. So the computer screen is the xy-plane
(vi) The z-axis is perpendicular to the computer screen
(vii) We already assumed that, the positive direction of the z-axis points towards us
(viii) So we can write:
The torque created by $\mathbf\small{\vec{F}_B}$ is perpendicular to the xy-plane and is directed towards the -ve side of the z axis
• Combining the magnitude and the direction, we can write:
The torque created by $\mathbf\small{\vec{F}_B}$ is: $\mathbf\small{[-d_2\times |\vec{F}_B|]\; \hat{k}}$ 
11. Next we find the torque created by $\mathbf\small{\vec{R}_C}$ about C
• But $\mathbf\small{\vec{R}_C}$ passes through C
• So the torque is $\mathbf\small{\vec{0}}$  
12. Now we can write a summary about the torques:
• The torque created by $\mathbf\small{\vec{F}_A}$:
$\mathbf\small{[d_1\times |\vec{F}_A|]\; \hat{k}}$
• The torque created by $\mathbf\small{\vec{F}_B}$:
$\mathbf\small{[-d_2\times |\vec{F}_B|]\; \hat{k}}$
• The torque created by $\mathbf\small{\vec{R}_C}$ is $\mathbf\small{\vec{0}}$
13. We want $\mathbf\small{\sum{\vec{\tau}_{z,ext}}=0}$
• Let us input the values:
$\mathbf\small{[(d_1\times |\vec{F}_A|)\hat{k}]-[(d_2\times |\vec{F}_B|)\hat{k}]+\vec{0}}$ must be equal to $\mathbf\small{\vec{0}}$
• That means, the following equation must be satisfied:
$\mathbf\small{[(d_1\times |\vec{F}_A|)-(d_2\times |\vec{F}_B|)]\hat{k}=\vec{0}}$
14. In the above equation, we have vectors on both sides. So their magnitudes must be equal
• Thus the condition becomes:
$\mathbf\small{[(d_1\times |\vec{F}_A|)-(d_2\times |\vec{F}_B|)]=0}$
15. The above equation can be rearranged as: $\mathbf\small{|\vec{F}_B|=\frac{|\vec{F}_A|d_1}{d_2}}$
• From the fig.7.97, we see that d1 is less than d2So $\mathbf\small{\frac{d_1}{d_2}}$ will be less than '1'
• Thus we find that: $\mathbf\small{\vec{F}_B}$ is less than $\mathbf\small{\vec{F}_A}$
16. Also we can write:
• The magnitude of the force applied at B should be equal to $\mathbf\small{\frac{|\vec{F}_A|d_1}{d_2}}$
• Then only we can achieve equilibrium for the rod AB
17. We can write the other way also:
• The magnitude of the force applied at A should be equal to $\mathbf\small{\frac{|\vec{F}_B|d_2}{d_1}}$
• Then only we can achieve equilibrium for the rod AB

Let us see a practical application of the equation:
In fig.7.99 below, a block of 10 kg hangs from the end A of the rod AB. When the rod is exactly horizontal, the block rests completely on the floor and the string is straight.
Working of the simple machine lever is based on rotational equilibrium
Fig.7.99
What force must be applied at B so that, the block just loses contact with the floor? 
[Take g = 10 ms-2]
Solution:
1. We have: $\mathbf\small{\vec{F}_A}$ = mg = 10 × 10 = 100 N
• Using equation in (16) above, we get:
$\mathbf\small{|\vec{F}_B|=\frac{|\vec{F}_A|d_1}{d_2}=\frac{100 \times 0.2}{0.8}=25\, \text{N}}$
2. 25 N corresponds to a mass of 2.5 kg
• So we are able to lift a large mass of 10 kg by a small mass of 2.5 kg

Note that, the 25 N corresponds to 'just losing contact between the block and the floor'
• If we lift the block further, the rod AB will no longer be horizontal
• As a result, the forces $\mathbf\small{\vec{F}_A}$ and $\mathbf\small{\vec{F}_B}$ will no longer be perpendicular to AB
• This situation is shown in fig.7.99(b)
■ If we want to keep the block stationary at a height much above the floor, what force must be applied at B?
Solution:
1. Fig.7.100(a) below shows a schematic diagram
Fig.7.100
AB makes an angle θ with the horizontal
2. We see that:
• $\mathbf\small{\vec{F}_A}$ and $\mathbf\small{\vec{F}_B}$ are vertical
    ♦ But $\mathbf\small{\vec{F}_A}$ is not perpendicular to AB
    ♦ $\mathbf\small{\vec{F}_B}$ is also not perpendicular to AB
3. So we resolve the two forces into components:
(i) $\mathbf\small{\vec{F}_A}$ has two components:
• One component is parallel to AB. It has magnitude: $\mathbf\small{|\vec{F}_A|\sin \theta}$ 
• The other component is perpendicular to AB. It has magnitude: $\mathbf\small{|\vec{F}_A|\cos \theta}$
(ii) $\mathbf\small{\vec{F}_B}$ has two components:
• One component is parallel to AB. It has magnitude: $\mathbf\small{|\vec{F}_B|\sin \theta}$ 
• The other component is perpendicular to AB. It has magnitude: $\mathbf\small{|\vec{F}_B|\cos \theta}$
• These components are shown in blue color in fig.b
4. The parallel components pass through C
• So they do not create any torque about C
• We need to consider the perpendicular components only
5. We can easily show the following two items:
(i) The torque created by the perpendicular component of $\mathbf\small{\vec{F}_A}$ is:
$\mathbf\small{[d_1\times |\vec{F}_A|\times \cos \theta]\; \hat{k}}$
(ii) The torque created by the perpendicular component of $\mathbf\small{\vec{F}_B}$ is:
$\mathbf\small{[-d_2\times |\vec{F}_B|\times \cos \theta]\; \hat{k}}$
(The reader may write the proofs in his/her own note books)
6. We want $\mathbf\small{\sum{\vec{\tau}_{z,ext}}=0}$
• Let us input the values:
$\mathbf\small{[(d_1\times |\vec{F}_A|\times \cos \theta)\hat{k}]-[(d_2\times |\vec{F}_B|\times \cos \theta)\hat{k}]}$ must be equal to $\mathbf\small{\vec{0}}$
• That means, the following equation must be satisfied:
$\mathbf\small{\left[(d_1\times |\vec{F}_A|\times \cos \theta)-(d_2\times |\vec{F}_B|\times \cos \theta)\right]\hat{k}=\vec{0}}$
14. In the above equation, we have vectors on both sides. So their magnitudes must be equal
• Thus the condition becomes:
$\mathbf\small{\left[(d_1\times |\vec{F}_A|\times \cos \theta)-(d_2\times |\vec{F}_B|\times \cos \theta)\right]=0}$
$\mathbf\small{\Rightarrow (d_1\times |\vec{F}_A|)-(d_2\times |\vec{F}_B|)=0}$
• This is the same result that we obtained earlier
15. The above equation can be rearranged as: $\mathbf\small{|\vec{F}_B|=\frac{|\vec{F}_A|d_1}{d_2}}$
16. Let us input the values:
$\mathbf\small{|\vec{F}_B|=\frac{100 \times 0.2}{0.8}=25\, \text{N}}$
• This is the same result that we obtained before

■ The rod AB in fig.7.99 is a lever
Let us see the important features of a lever:
(i) An ideal lever is a light (ie. of negligible mass) rod pivoted at a point along its length
(ii) The pivot point is called the fulcrum
(iii) The force $\mathbf\small{\vec{F}_A}$ is usually some weight to be lifted
• This $\mathbf\small{\vec{F}_A}$ is called the load
• Distance (d1) of load from the fulcrum is called load arm
(iv) The force $\mathbf\small{\vec{F}_B}$ applied to lift the load
• This $\mathbf\small{\vec{F}_B}$ is called the effort
• Distance (d2) of effort from the fulcrum is called effort arm
(v) Consider the expression '$\mathbf\small{(d_1\times |\vec{F}_A|)-(d_2\times |\vec{F}_B|)=0}$' that we derived in (14) above
• It can be rearranged as: $\mathbf\small{d_1\times |\vec{F}_A|=d_2\times |\vec{F}_B|}$
• That is: Load arm × Load = Effort arm × Effort
• This is the mathematical expression of the Principle of moments for a lever
(vi) The ratio $\mathbf\small{\frac{Load}{Effort}}$ is called Mechanical advantage
• From (v) above, we get: mechanical advantage = $\mathbf\small{\frac{\text{Effort arm}}{\text{Load arm}}}$
(vii) If $\mathbf\small{\frac{\text{Effort arm}}{\text{Load arm}}}$ is greater than '1', we will be able to lift a larger load by a smaller effort
(viii) A see saw at the children’s park is an example of a lever
Some other examples are:
Cutting pliers
Wheel barrow
Nut cracker etc.

In the next section, we will see Center of gravity

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