A solid sphere of radius 3 cm and mass of 100 g rolls down a 30 degree incline from a height of 1 m. A hoop of the same mass and radius rolls down the same incline from the same height. What is the linear velocity of each body at the bottom of the incline? From what height would the body with the larger moment of inertia have to roll down from so that the two linear velocities at the bottom of the incline would be the same?

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Solid spheres will roll faster than hoops.

To find the linear velocity of each body at the bottom of the incline, we can use the principle of conservation of energy. The potential energy at the top of the incline is converted into kinetic energy at the bottom.

For the solid sphere:

1. Calculate the potential energy at the top of the incline.
Potential energy = mass * gravity * height
In this case, the mass of the sphere is given as 100 g, which is 0.1 kg, gravity is 9.8 m/s^2, and the height is 1 m.
Potential energy = 0.1 kg * 9.8 m/s^2 * 1 m = 0.98 J

2. Calculate the kinetic energy at the bottom of the incline.
Since the sphere is rolling without slipping, the total kinetic energy is the sum of translational and rotational kinetic energies.
Translational kinetic energy = (1/2) * mass * velocity^2
Rotational kinetic energy = (1/2) * moment of inertia * angular velocity^2
For a solid sphere:
Moment of inertia = (2/5) * mass * radius^2
In this case, the radius of the sphere is given as 3 cm, which is 0.03 m.
Moment of inertia = (2/5) * 0.1 kg * (0.03 m)^2 = 0.00036 kg*m^2

3. Equate the initial potential energy to the final kinetic energy to find the linear velocity.
Potential energy = Translational kinetic energy + Rotational kinetic energy

0.98 J = (1/2) * 0.1 kg * velocity^2 + (1/2) * 0.00036 kg*m^2 * (velocity / radius)^2

Solve this equation to find the linear velocity of the solid sphere at the bottom of the incline.

Repeat the same steps for the hoop using its respective moment of inertia formula:
Moment of inertia for a hoop = mass * radius^2

To find the height from which the body with the larger moment of inertia needs to roll down to have the same linear velocity as the solid sphere, the principle of conservation of energy still applies. We need to equate their kinetic energies at the bottom.

0.5 * mass * v_sphere^2 = 0.5 * mass * v_hoop^2

Since the mass is the same in this case, we can cancel it out.

v_sphere^2 = v_hoop^2

Solve this equation to find the height from which the body with the larger moment of inertia needs to roll down.