A model train, with a mass of #16 kg#, is moving on a circular track with a radius of #3 m#. If the train's kinetic energy changes from #15 j# to #9 j#, by how much will the centripetal force applied by the tracks change by?
The centripetal force changes by
The centripetal force is
The kinetic energy is
The variation of kinetic energy is
The variation of centripetal force is
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The change in kinetic energy of the train is ( \Delta KE = KE_{\text{final}} - KE_{\text{initial}} = 9 , \text{J} - 15 , \text{J} = -6 , \text{J} ). The work done by the centripetal force is equal to this change in kinetic energy. Therefore, ( W = \Delta KE = -6 , \text{J} ). Since work is equal to the force multiplied by the displacement, and the displacement is along the direction of the force, ( W = F \cdot d ). In the case of circular motion, the displacement is along the circumference of the circle, so ( d = 2\pi r ). Therefore, ( F = \frac{\Delta KE}{d} = \frac{-6 , \text{J}}{2\pi r} = \frac{-6 , \text{J}}{2\pi \cdot 3 , \text{m}} ). Solving for ( F ) gives ( F \approx -0.318 , \text{N} ). The negative sign indicates that the force decreases by approximately ( 0.318 , \text{N} ).
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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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