A non-homogeneous trunk, 5 meters long and 100 kilograms mass is in equilibrium and is hung at the ends by two strings; One form an angle of 37 degrees and the other an angle of 53 degrees, both with the horizontal.
Determine the location of the center of mass of the trunk and the tension of both strings.
I need helo, I already did the free boda diagrama but I don't know how TO calcula te the CM as it is not a regular object. Thanks.
Total force up
=100(9.81) = T1 sin37 + T2 sin 53
Total horizontal force
= 0 = T1 cos 37 - T2 cos 53
use that to get T1 and T2
now for example moments about left end
T2 sin 53 *5 = 100*9.81* X
You could have done that, patience :)
To determine the location of the center of mass (CM) of the non-homogeneous trunk, you can use the concept of the weighted average. The CM represents the point where the entire mass of the trunk can be considered to be concentrated.
Here's how you can find the CM:
1. Divide the trunk into small segments along its length. You can consider each segment as a point mass.
2. Assign each segment a mass dm (a small mass element). Since the trunk is non-homogeneous, the mass will vary as you move along its length.
3. Choose a coordinate system with the origin at one end of the trunk. Let's assume the positive x-axis points towards the other end of the trunk.
4. Express the position of each mass element dm as a distance x from the origin.
5. Use the given length of the trunk (5 meters) to set up a variable, such as 'a', which represents the position of the CM from the origin.
6. Integrate the positions of all the mass elements (dm) in terms of their distances from the origin (x) using the formula for the center of mass:
∫x dm = a
This integral integrates over the entire length of the trunk.
7. Evaluate the integral using the known variations in mass. The specific mathematical form of the mass distribution will determine how the integral is evaluated. You may need to use calculus or appropriate techniques to solve the integral.
Once you have determined the location of the CM (a), you can proceed to calculate the tension in each string using the angles provided.
Let me know if you would like further assistance with the trigonometry involved in determining the tensions in the strings.