Translational Kinetic Energies + Plane

In summary, the question asks to find the ratio of the translational kinetic energies of a ring, a coin, and a solid sphere at the bottom of an inclined plane, assuming pure rolling without any slipping. The correct approach is to obtain the kinetic energy in terms of potential energy for each object and then take the ratios of the kinetic energies. The potential energy for each object is the same since they all start at the same height and have the same mass.
  • #1
Kishor Bhat
22
0

Homework Statement


Find the ratio of the translational kinetic energies of a ring, a coin, and a solid sphere at the bottom of an inclined plane. The bodies have been released from rest at the top. Assume pure rolling without any slipping.


The Attempt at a Solution



Well, I'm really not sure. Since T.K.=1/2 m*v^2, we need to look at linear velocities v at the bottom of the plane for each object. Obviously the rotation will influence this, and I've tried obtaining final ω and using v=ωr at the bottom, but I'm not getting the answer. Which, by the way, is 21:28:30.
 
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  • #2
Perhaps you are confusing the linear velocity from v=ωr, which is the instantaneous velocity of the edge of the rolling object, with the linear velocity of the center of mass of the object.

Beside that, are you using the correct moments of inertia for the different objects? I'm assuming the masses and radii are all the same.
 
  • #3
The MI's are correct, and yes, they are all of same mass and radius. One question: can we find the acceleration of c.o.m and then use the equations of motion to find final velocity? I haven't tried that.
 
  • #4
PICsmith said:
Perhaps you are confusing the linear velocity from v=ωr, which is the instantaneous velocity of the edge of the rolling object, with the linear velocity of the center of mass of the object.

Nevermind this statement...while yes they are different things, in this case they are the same velocity (the center of mass velocity in the frame at rest w.r.t. the ramp is of course the same as the linear velocity of the edge in the co-moving frame, since there is no slipping).

You don't have to calculate any velocites. Here's how you go about it. For example, for the ring you have KE = PE-RE = PE-(1/2)Iω^2 = PE-(1/2)mv^2 = PE-KE, and so you have KE=PE/2. Obtain the KE in terms of PE for the other two objects, and then you can take the ratios between them.

Edit: Keep in mind that the PE for each object is the same since they all start at the same height and have the same mass.
 
Last edited:
  • #5
Gracias.
 

What is translational kinetic energy?

Translational kinetic energy is the energy an object possesses due to its motion in a straight line. It is a form of kinetic energy that is dependent on the mass and velocity of the object.

How is translational kinetic energy calculated?

The formula for translational kinetic energy is KE = 1/2 * mv^2, where m is the mass of the object and v is its velocity. This formula is derived from the work-energy theorem and is measured in joules (J).

What is the relationship between translational kinetic energy and plane motion?

Plane motion refers to the movement of an object in two dimensions, typically in a straight line or along a curved path. Translational kinetic energy is a type of energy that describes the motion of an object in a straight line, making it directly related to plane motion.

How does the angle of inclination affect translational kinetic energy in a plane?

The angle of inclination, or slope, of a plane affects the translational kinetic energy of an object by changing its velocity. If the angle of inclination is steeper, the object will accelerate and have a higher velocity, increasing its translational kinetic energy. On the other hand, a shallower angle of inclination will result in a slower velocity and lower translational kinetic energy.

What are some real-world applications of translational kinetic energy in plane motion?

Translational kinetic energy in plane motion is used in various fields, such as aviation, engineering, and sports. For example, airplanes rely on translational kinetic energy to stay in flight, while roller coasters use it to propel riders along their tracks. In sports, athletes use their translational kinetic energy to run, jump, and throw objects.

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