Rotational kinetic energy problem

You'll find a clue there that will help you to find the amount of translational kinetic energy.In summary, the problem involves a solid sphere with a mass of 8.2kg and radius of 10cm sliding along a frictionless surface with a speed of 5.4m/s. The sphere is also spinning, with 0.31 of its total kinetic energy in translational motion. The goal is to find the speed of the sphere's spinning. To do so, we need to calculate the rotational kinetic energy, which can be found using the equation KAe = (1/2)(I)(W)^2. However, the problem statement does not provide the rotational kinetic energy, so we must first find the transl
  • #1
blackbyron
48
0

Homework Statement


A solid sphere with a mass 8.2kg and radius 10cm is sliding along a frictionless surface with a speed 5.4m/s while at the same time spinning. The sphere has 0.31 of its total kinetic energy in translational motion. How fast is the sphere spinning?


Homework Equations



KAe = (1/2)(I)(W)^2


The Attempt at a Solution



The problem is that the rotational kinetic energy is unknown, but I found that the inertia is .0328kgm^2

How do I find the rotation kinetic energy so I can solve for w?
 
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  • #2
In the relevant equations, you have left out the formula for calculating the kinetic energy due to translation of the sphere at the given speed.
 
  • #3
SteamKing said:
In the relevant equations, you have left out the formula for calculating the kinetic energy due to translation of the sphere at the given speed.

You mean this ?

Translation:

Ke = (1/2)(m)(v^2)
 
  • #4
Ya. Now reread your problem statement very carefully.
 
  • #5


To find the rotational kinetic energy, you can use the equation Krot = (1/2)(I)(ω)^2, where I is the moment of inertia and ω is the angular velocity. In this problem, the moment of inertia of a solid sphere is (2/5)MR^2, where M is the mass and R is the radius. So, in this case, I = (2/5)(8.2kg)(0.1m)^2 = 0.0328 kgm^2.

Since the problem states that the sphere has 0.31 of its total kinetic energy in translational motion, we can write the equation: Ktrans = (0.31)(Ktotal). Since the total kinetic energy is equal to the sum of translational and rotational kinetic energy, we can write: Ktotal = Ktrans + Krot.

Plugging in the known values, we get: (0.31)(Ktotal) = (0.31)(0.5)(8.2kg)(5.4m/s)^2. Solving for Ktotal, we get Ktotal = 127.73J.

Now, we can plug this value into the equation for rotational kinetic energy and solve for ω: 127.73J = (0.5)(0.0328kgm^2)(ω)^2. Solving for ω, we get ω = 15.58 rad/s.

Therefore, the sphere is spinning at a speed of 15.58 rad/s.
 

1. What is rotational kinetic energy and how is it different from linear kinetic energy?

Rotational kinetic energy is the energy an object possesses due to its rotation around an axis. It is different from linear kinetic energy, which is the energy an object possesses due to its linear motion. Rotational kinetic energy depends on an object's moment of inertia, angular velocity, and angular momentum, while linear kinetic energy depends on an object's mass, velocity, and momentum.

2. How is rotational kinetic energy calculated?

The formula for calculating rotational kinetic energy is Erot = 1/2 * I * ω2, where Erot is the rotational kinetic energy, I is the moment of inertia, and ω is the angular velocity.

3. What is the relationship between rotational kinetic energy and rotational inertia?

Rotational kinetic energy is directly proportional to the moment of inertia, which is a measure of an object's resistance to changes in its rotational motion. This means that as the moment of inertia increases, the rotational kinetic energy also increases.

4. Can rotational kinetic energy be converted into other forms of energy?

Yes, rotational kinetic energy can be converted into other forms of energy, such as heat and sound. This is because energy is conserved and can be transferred from one form to another.

5. How does rotational kinetic energy affect the stability of an object?

The greater the rotational kinetic energy of an object, the less stable it becomes. This is because a higher rotational kinetic energy means a higher angular velocity, which can cause the object to become unbalanced and potentially fall over. On the other hand, a lower rotational kinetic energy means a lower angular velocity and a more stable object.

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