# A model train, with a mass of #4 kg#, is moving on a circular track with a radius of #3 m#. If the train's kinetic energy changes from #4 j# to #15 j#, by how much will the centripetal force applied by the tracks change by?

we put this value in centripetal force change equation we get

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To find the change in the centripetal force applied by the tracks, we first need to calculate the initial and final speeds of the train.

The initial kinetic energy ( K_{initial} ) is given as 4 J, and the final kinetic energy ( K_{final} ) is given as 15 J.

The formula for kinetic energy is ( K = \frac{1}{2} mv^2 ), where ( m ) is the mass of the object and ( v ) is its velocity.

First, we find the initial velocity ( v_{initial} ) using the initial kinetic energy, and then we find the final velocity ( v_{final} ) using the final kinetic energy.

[ K_{initial} = \frac{1}{2} m v_{initial}^2 ] [ 4 = \frac{1}{2} \times 4 \times v_{initial}^2 ] [ v_{initial}^2 = 2 ] [ v_{initial} = \sqrt{2} ]

Similarly, [ K_{final} = \frac{1}{2} m v_{final}^2 ] [ 15 = \frac{1}{2} \times 4 \times v_{final}^2 ] [ v_{final}^2 = \frac{15 \times 2}{4} ] [ v_{final}^2 = 7.5 ] [ v_{final} = \sqrt{7.5} ]

Now, we use the formula for centripetal force ( F_c = \frac{mv^2}{r} ) to find the initial and final centripetal forces.

For the initial situation: [ F_{c_{initial}} = \frac{4 \times (2)}{3} ] [ F_{c_{initial}} = \frac{8}{3} ]

For the final situation: [ F_{c_{final}} = \frac{4 \times (7.5)}{3} ] [ F_{c_{final}} = 10 ]

Now, we find the change in centripetal force: [ \Delta F_c = F_{c_{final}} - F_{c_{initial}} ] [ \Delta F_c = 10 - \frac{8}{3} ] [ \Delta F_c = \frac{30}{3} - \frac{8}{3} ] [ \Delta F_c = \frac{22}{3} ]

So, the change in the centripetal force applied by the tracks is ( \frac{22}{3} ) 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.

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