Two spheres of mass 2 kg and 3 kg are attached to the ends of a light rigid rod of length 1.0 m. What is the moment of inertia of the system about an axis perpendicular to the rod and
(i) passes through the centre of the rod
(ii) passes through the 2 kg sphere
In the amusement park ride known as Magic Mountain Superman, powerful magnets accelerate a car and its riders from rest to 36.1 m/s in a time of 7.70 s. The mass of the car and riders is 5.62 × 103 kg. Find the average net force exerted on the car and riders by the magnets ?
To find the moment of inertia of the system, we need to consider the moment of inertia of each sphere individually and the moment of inertia of the rod.
(i) To find the moment of inertia about an axis passing through the center of the rod:
First, we need to find the moment of inertia of each sphere. The moment of inertia of a sphere about an axis passing through its center can be calculated using the formula:
I_sphere = (2/5) * m * r^2
Where m is the mass of the sphere and r is the radius of the sphere.
For the 2 kg sphere:
I_2kg = (2/5) * 2 kg * (radius of the sphere)^2
Now, to find the moment of inertia of the rod, we use the formula for the moment of inertia of a rod rotating about its center:
I_rod = (1/12) * m * L^2
Where m is the mass of the rod and L is the length of the rod.
For the rod:
I_rod = (1/12) * (mass of the rod) * (length of the rod)^2
Finally, we can add the moment of inertia of each sphere and the moment of inertia of the rod to find the total moment of inertia of the system:
Total moment of inertia = I_2kg + I_rod
(ii) To find the moment of inertia about an axis passing through the 2 kg sphere:
In this case, since the axis passes through the 2 kg sphere, the moment of inertia of the 3 kg sphere is not considered.
We only need to find the moment of inertia of the 2 kg sphere about an axis passing through it. This can be calculated using the formula stated earlier:
I_2kg_sphere = (2/5) * (mass of the sphere) * (radius of the sphere)^2
So, the moment of inertia of the system about an axis passing through the 2 kg sphere is equal to the moment of inertia of the 2 kg sphere:
Total moment of inertia = I_2kg_sphere
Remember to substitute the appropriate values for mass and length in the formulas to calculate the moment of inertia.