The rigid body shown in Figure 10-66 (it is a triangle with the upper mass = to M and the bottom two corners/masses equal to 2M. The bottom side length equals .70 cm with point P in the middle of the side and the other two sides equal .55 cm)consists of three particles connected by massless rods. It is to be rotated about an axis perpendicular to its plane through point P. If M = 0.40 kg, a = 35 cm, and b = 55 cm, how much work is required to take the body from rest to an angular speed of 5.0 rad/s?
*My work so far*
I believe I need to use:
W = (1/2)(I)(ang speed)^2
I am not sure how to find I though. Would I use I = I1 + I2 + I3 and calculate I for each of the 3 masses?
Yes, use the addition formula to get the moment of inertia. In each case, the distance is the distance of the mass from point P.
To calculate the moment of inertia for each mass, you need to use the formulas for the moment of inertia of different shapes.
1. For the upper mass (M), which is located at the top of the triangle:
The moment of inertia (I1) for a point mass rotating about an axis perpendicular to its plane and passing through its center of mass is given by:
I1 = M * r1^2
Here, r1 is the distance between point P and the mass. In this case, it is equal to half the length of the top side of the triangle since the mass is located at the center of that side.
2. For the two bottom masses (2M), which are located at the bottom corners of the triangle:
The moment of inertia (I2 and I3) for point masses rotating about an axis perpendicular to its plane and passing through an end is given by:
I2 = I3 = 2M * r2^2
Here, r2 is the distance between point P and one of the bottom masses. In this case, it is equal to half the length of the bottom side of the triangle since the mass is located at the end of that side.
Now that you have the individual moments of inertia, you can calculate the total moment of inertia (I) using the addition formula:
I = I1 + I2 + I3
Substitute the given values into the formulas to calculate the moment of inertia for each mass, and then add them up to find the total moment of inertia (I).
Once you have the moment of inertia (I), you can use it in the work formula:
W = (1/2) * I * (angular speed)^2
Substitute the given values of I and the angular speed to calculate the work required to take the body from rest to an angular speed of 5.0 rad/s.