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Centre of mass

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Summary

Centre of mass

In a nutshell

Objects can be non-uniform, which means the centre of mass is not necessarily at the midpoint. Using forces and moments you can find the centre of mass.



Centre of mass

With a non-uniform rod, you may have to find the position of its centre of mass. The centre of mass can be found by considering the moment due to the weight of the rod. By taking moments about a point, you can create an equation to find the distance of the centre of mass from the point.


Example 1

A non-uniform rod ABABAB is 4m4m4m long and has a weight of 25N25N25N. It rests in a horizontal equilibrium on supports CCC and DDD, where AC=1mAC = 1mAC=1m and AD=3mAD = 3mAD=3m. The magnitude of the reaction at CCC is four times the magnitude of the reaction at DDD. Find the distance of the centre of mass of the rod from AAA.

Maths; Moments; KS5 Year 13; Centre of mass


Find the magnitude of the reactions:

Take the reaction at DDD to be RRR, therefore the reaction at CCC is 4R4R4R. Resolve vertically:


​4R+R=255R=25R=5\begin{aligned}4R+R&=25\\ 5R&=25\\ R&=5\end{aligned}4R+R5RR​=25=25=5​​​


Therefore, the reaction at CCC is 20N20N20N and the reaction at DDD is 5N5N5N. Take moments about point A, supposing the centre of mass of the rod acts a point xmxmxm from AAA:


​25x=20(1)+5(3)25x=20+1525x=35x=3525x=1.4\begin{aligned}25x&=20(1)+5(3)\\ 25x&=20+15\\25x&=35\\ x&=\dfrac{35}{25}\\ x&=1.4\end{aligned}25x25x25xxx​=20(1)+5(3)=20+15=35=2535​=1.4​​​


Therefore, the centre of the mass of the rod is 1.4m‾\underline{1.4m}1.4m​ from AAA.


Example 2

Two people are sitting on a seesaw which is modelled as a non-uniform plank ABABAB with a mass of 15kg15kg15kg and a length of 4m4m4m. The plank rests on a pivot PPP​ at the midpoint of ABABAB. The centre of mass of the plank is at a point CCC, where AC=1.5mAC = 1.5mAC=1.5m​ Person FFF has a mass of 20kg20kg20kg and sits at point AAA. Person GGG has a mass of 35kg35kg35kg and is sitting on the other side of the plank. Where on the plank does GGG have to sit for the plank to be horizontal? (G can be moved from the end of the plank B towards the pivot).


Maths; Moments; KS5 Year 13; Centre of mass


Take moments about the pivot PPP. Take person GGG's distance from the midpoint to be xmxmxm:


​(20g×2)+(15g×0.5)=35g×x40g+7.5g=35xg47.5=35x1.36=x\begin{aligned}(20g\times 2)+(15g\times 0.5)&=35g\times x\\ 40g+7.5g&=35xg\\47.5&=35x\\ 1.36&=x\end{aligned}(20g×2)+(15g×0.5)40g+7.5g47.51.36​=35g×x=35xg=35x=x​​​



Therefore for the plank to be horizontal, person GGG must be sitting 1.36m‾\underline{1.36m}1.36m​ from the pivot.


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FAQs - Frequently Asked Questions

What does a non-uniform rod mean?

The centre of mass is not necessarily at the midpoint.

What is the difference between the mass and weight af an object?

Mass is measured in kg, and weight is mass times gravity measured in N.

How do you use moments to find the centre of mass?

By taking moments about a point, you can create an equation to find the distance of the centre of mass from the point.

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