Calculating Work and Kinetic Energy in Uniform Circular Motion

In summary, a particle of mass m is moving in uniform circular motion with speed v0 and radius R0. By slowly reducing the radius to R1, the work done by the string force can be shown to be equal to the change in kinetic energy. This is derived from the equations for tension, kinetic energy, and angular momentum conservation. The integral of the centripetal force multiplied by the distance moved gives the change in kinetic energy, with the same expression obtained from the difference in kinetic energy equation.
  • #1
captain.joco
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Homework Statement



A particle of mass m is tied to a string which goes through a hole in a smooth horizontal table. The particle moves in uniform circular motion with speed v0. The radius of the cricle is R0. By puling the string very slowly, the radius of the circular motion is reduced tao R1.
Show that the work done by the string force is equal to the change of Kinetic energy.

Homework Equations


F=Tension(T) = mv^2 /R
K ( kinetic energy ) =1/2mv^2
K1-K0 ( final - initial kinetic energy ) = 1/2m ( v1^2 - v0^2 )
where v1 = R0*v0/R1 ( using angular momentum conservation ).


The Attempt at a Solution


W=∆K ( work energy theorem )
W = ∫Fdr , where F = m*v*v / r ( centripetal force )
I think I need some substitution to get the integral to give the ∆E...
 
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  • #2
With the integrand of F = m*v*v/r multiplied by dr, one has units of kg m2/s2 (assuming SI) which is units of kinetic energy.
 
  • #3
I know that the units are consistent. I have found the change in Kinetic energy, which is deltaK = 1/2 *v0^2 * [( R0^2 / R1^2 ) - 1 ].
But when I integrate mv1^2/R1 dR ( between R0 and R1 ), I get the very same expression but without the 1/2 factor at the start.. What am I doing wrong?
 
  • #4
What's the force as a function of R?
 
  • #5
I solved the integral and I got the same expression as in for the difference in kinetic energy, so it's got to be right. Thanks for Your help!
 

What is work?

Work is defined as the transfer of energy from one system to another. In scientific terms, work is done when a force is applied to an object and the object is moved in the direction of the force.

What is force?

Force is a physical quantity that is used to describe the strength or magnitude of an interaction between two objects. It is a vector quantity, meaning it has both magnitude and direction.

What is energy?

Energy is the ability to do work. It is a physical quantity that can be transferred between objects or converted into different forms, such as mechanical, thermal, or electrical energy.

What is the relationship between work, force, and energy?

Work and energy are directly related, as work is the transfer of energy from one object to another. Force is also a key factor in work, as it is the force applied that results in the transfer of energy.

How is work, force, and energy measured?

Work is measured in joules (J) in the SI system, force is measured in newtons (N), and energy is also measured in joules (J). Different methods and instruments are used to measure these quantities, such as force gauges and energy meters.

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