A solid sphere is rotating about an axis through its center at a constant rotation rate. Another hollow sphere of the same mass and radius is rotating about its axis through the center at the same rotation rate. Which sphere has a greater rotational kinetic energy?Group of answer choicesThere is not enough information to determineTheir kinetic energies would be equalThe solid sphereThe hollow sphere
Question
A solid sphere is rotating about an axis through its center at a constant rotation rate. Another hollow sphere of the same mass and radius is rotating about its axis through the center at the same rotation rate. Which sphere has a greater rotational kinetic energy?Group of answer choicesThere is not enough information to determineTheir kinetic energies would be equalThe solid sphereThe hollow sphere
Solution 1
To determine which sphere has a greater rotational kinetic energy, we need to consider the formula for rotational kinetic energy. The rotational kinetic energy (KE) of an object is given by the equation KE = (1/2) I ω^2, where I is the moment of inertia and ω is the angular velocity.
Since both spheres have the same mass and rotation rate, we can assume that their angular velocities (ω) are equal. However, the moment of inertia (I) depends on the distribution of mass within the object.
For a solid sphere, the moment of inertia is given by I = (2/5) m r^2, where m is the mass and r is the radius. For a hollow sphere, the moment of inertia is given by I = (2/3) m r^2.
Comparing the two equations, we can see that the moment of inertia for a solid sphere is smaller than that of a hollow sphere. Therefore, the solid sphere will have a greater rotational kinetic energy.
Hence, the answer is: The solid sphere has a greater rotational kinetic energy.
Solution 2
To determine which sphere has a greater rotational kinetic energy, we need to consider the formula for rotational kinetic energy. The rotational kinetic energy (KE) of an object is given by the equation KE = (1/2) I ω^2, where I is the moment of inertia and ω is the angular velocity.
Since both spheres have the same mass and rotation rate, we can assume that their angular velocities (ω) are equal. However, the moment of inertia (I) will be different for the solid and hollow spheres.
The moment of inertia depends on the mass distribution of the object. For a solid sphere, the moment of inertia is given by I = (2/5) m r^2, where m is the mass and r is the radius. For a hollow sphere, the moment of inertia is given by I = (2/3) m r^2.
Comparing the two equations, we can see that the moment of inertia for the solid sphere is smaller than that of the hollow sphere. Therefore, the solid sphere will have a greater rotational kinetic energy.
Hence, the answer is: The solid sphere has a greater rotational kinetic energy.
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Multiple Choice QuestionWhat will be the kinetic energy of a ball if its mass is 0.145 kg and it travels at a velocity of 9 meters per second?A.4.9 joulesB.5.9 joulesC.6.9 joulesD.7.9 joules
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