An object is formed by attaching a uniform, thin rod with a mass of mr = 6.54 kg and length L = 5.12 m to a uniform sphere with mass ms = 32.7 kg and radius R = 1.28 m. Note ms = 5mr and L = 4R.
1)What is the moment of inertia of the object about an axis at the left end of the rod?
1417.969664
2)If the object is fixed at the left end of the rod, what is the angular acceleration if a force F = 467 N is exerted perpendicular to the rod at the center of the rod?
.8431209992
3)What is the moment of inertia of the object about an axis at the center of mass of the object? (Note: the center of mass can be calculated to be located at a point halfway between the center of the sphere and the left edge of the sphere.)
4)If the object is fixed at the center of mass, what is the angular acceleration if a force F = 467 N is exerted parallel to the rod at the end of rod?
0
5)What is the moment of inertia of the object about an axis at the right edge of the sphere?
260.734976
All above answers are correct, Ive been having troubles with problem 3 for several hours I also talk to a bunch of TAs and even they couldn't help me out. If any one could help me out I would appreciate it. Thanks.
I(end) = I(center of mass) + M(total)D^2
Solve for I(CM) = I(end) - M(total)D^2
M(total) = m(sphere) + m(rod)
D = Distance from end of rod to CM = L + R/2
I(end) = I(rod) + I(sphere)
I(rod) = 1/3(mass of rod)(Length of rod)^2
I(end, sphere) = 2/5(mass of sphere)(Radius)^2 + (mass of sphere)D^2
Here, D = distance from CM to left axis = L + R
Certainly! I'll be happy to help you with problem 3.
To find the moment of inertia of the object about an axis at the center of mass, we need to consider the individual moments of inertia of the rod and the sphere about their respective centers of mass and then use the parallel axis theorem.
First, let's find the moment of inertia of the rod about its center of mass, using the formula for the moment of inertia of a thin rod about its center:
I_rod = (1/12) * m_rod * L^2
where m_rod is the mass of the rod and L is its length.
Substituting the given values, we have:
I_rod = (1/12) * 6.54 kg * (5.12 m)^2
I_rod = 83.39946626 kg * m^2
Next, let's find the moment of inertia of the sphere about its center of mass, using the formula for the moment of inertia of a sphere:
I_sphere = (2/5) * m_sphere * R^2
where m_sphere is the mass of the sphere and R is its radius.
Substituting the given values, we have:
I_sphere = (2/5) * 32.7 kg * (1.28 m)^2
I_sphere = 66.127872 kg * m^2
Now, let's find the distance between the center of the sphere and the left edge of the sphere, which is half the radius:
Distance = R/2
Distance = (1.28 m)/2
Distance = 0.64 m
Finally, we can use the parallel axis theorem to find the moment of inertia of the object about an axis at the center of mass:
I_center = I_rod + I_sphere + m_rod * Distance^2
Substituting the values we have calculated, we get:
I_center = 83.39946626 kg * m^2 + 66.127872 kg * m^2 + 6.54 kg * (0.64 m)^2
I_center = 83.39946626 kg * m^2 + 66.127872 kg * m^2 + 6.54 kg * 0.4096 m^2
I_center = 144.89933826 kg * m^2 + 27.2254464 kg * m^2
I_center = 172.12478466 kg * m^2
Therefore, the moment of inertia of the object about an axis at the center of mass is approximately 172.12478466 kg * m^2.
Sure, I can help you with problem 3.
To find the moment of inertia of the object about an axis at the center of mass, we need to consider the individual moments of inertia of the rod and the sphere and then use the parallel axis theorem to add them up.
The moment of inertia of the rod about its center is given by the formula:
I_rod = (1/12) * m_rod * L^2
Where m_rod is the mass of the rod and L is its length. Substituting the given values, we have:
I_rod = (1/12) * 6.54 kg * (5.12 m)^2
Next, let's find the moment of inertia of the sphere about its center. The moment of inertia of a sphere is given by the formula:
I_sphere = (2/5) * m_sphere * R^2
Where m_sphere is the mass of the sphere and R is its radius. Substituting the given values, we have:
I_sphere = (2/5) * 32.7 kg * (1.28 m)^2
Now, using the parallel axis theorem, we can add the moments of inertia of the rod and the sphere about their individual centers to get the moment of inertia about the center of mass:
I_center = I_rod + I_sphere
Finally, we divide this moment of inertia by the total mass of the object to get the moment of inertia per unit mass:
I_center = (I_rod + I_sphere) / (m_rod + m_sphere)
Substituting the previously calculated values, we have:
I_center = (I_rod + I_sphere) / (6.54 kg + 32.7 kg)
I_center = (1/12) * 6.54 kg * (5.12 m)^2 + (2/5) * 32.7 kg * (1.28 m)^2 / (6.54 kg + 32.7 kg)
Calculating this expression gives us the moment of inertia of the object about an axis at the center of mass.