# A model train with a mass of #4 kg# is moving along a track at #3 (cm)/s#. If the curvature of the track changes from a radius of #4 cm# to #27 cm#, by how much must the centripetal force applied by the tracks change?

The centripetal force changes by

The centripetal force is

The variation in centripetal force is

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To calculate the change in centripetal force, use the formula:

[ F_c = \frac{{m \cdot v^2}}{r} ]

Where:

- ( F_c ) is the centripetal force,
- ( m ) is the mass of the train (4 kg),
- ( v ) is the velocity of the train (3 cm/s),
- ( r ) is the radius of curvature of the track.

First, calculate the initial centripetal force when the radius is 4 cm: [ F_{c1} = \frac{{4 , \text{kg} \cdot (3 , \text{cm/s})^2}}{{4 , \text{cm}}} ]

Then, calculate the final centripetal force when the radius is 27 cm: [ F_{c2} = \frac{{4 , \text{kg} \cdot (3 , \text{cm/s})^2}}{{27 , \text{cm}}} ]

Finally, find the difference between the initial and final centripetal forces: [ \Delta F_c = F_{c2} - F_{c1} ]

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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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