# A model train with a mass of #5 kg# is moving along a track at #14 (cm)/s#. If the curvature of the track changes from a radius of #84 cm# to #28 cm#, by how much must the centripetal force applied by the tracks change?

The centripetal force changes by

Centripetal force is what

The centripetal force fluctuation is

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The centripetal force applied by the tracks must change according to the difference in curvature of the track. The centripetal force is given by the formula:

( F = \frac{mv^2}{r} )

where:

- ( F ) is the centripetal force,
- ( m ) is the mass of the train (5 kg),
- ( v ) is the velocity of the train (14 cm/s), and
- ( r ) is the radius of curvature of the track.

Initially, the radius of curvature of the track is 84 cm. Substituting the values into the formula:

( F_1 = \frac{5 \times (14)^2}{84} )

( F_1 = \frac{5 \times 196}{84} )

( F_1 = \frac{980}{84} )

( F_1 = 11.67 ) N

Now, the radius of curvature changes to 28 cm. Substituting the values into the formula:

( F_2 = \frac{5 \times (14)^2}{28} )

( F_2 = \frac{5 \times 196}{28} )

( F_2 = \frac{980}{28} )

( F_2 = 35 ) N

The change in centripetal force applied by the tracks is given by the difference between the forces:

( \Delta F = F_2 - F_1 )

( \Delta F = 35 - 11.67 )

( \Delta F \approx 23.33 ) N

Therefore, the centripetal force applied by the tracks must increase by approximately 23.33 N when the curvature of the track changes from a radius of 84 cm to 28 cm.

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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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