# A model train with a mass of #4 kg# is moving along a track at #21 (cm)/s#. If the curvature of the track changes from a radius of #84 cm# to #140 cm#, by how much must the centripetal force applied by the tracks change?

#=21000-12600 " dyne" =8400" dyne" =0.084 N#

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

[F_c = \frac{mv^2}{r}]

Where:

- (F_c) is the centripetal force
- (m) is the mass of the train
- (v) is the velocity of the train
- (r) is the radius of the curvature of the track

Substituting the given values into the formula:

Initial radius, (r_1 = 84 , \text{cm}) Initial centripetal force, (F_{c1} = \frac{4 , \text{kg} \times (21 , \text{cm/s})^2}{84 , \text{cm}})

Final radius, (r_2 = 140 , \text{cm}) Final centripetal force, (F_{c2} = \frac{4 , \text{kg} \times (21 , \text{cm/s})^2}{140 , \text{cm}})

Change in centripetal force, (\Delta F_c = F_{c2} - F_{c1})

[ \Delta F_c = \left( \frac{4 , \text{kg} \times (21 , \text{cm/s})^2}{140 , \text{cm}} \right) - \left( \frac{4 , \text{kg} \times (21 , \text{cm/s})^2}{84 , \text{cm}} \right) ]

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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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- Two objects have masses of #35 MG# and #21 MG#. How much does the gravitational potential energy between the objects change if the distance between them changes from #8 m# to #3 m#?
- A model train with a mass of #3 kg# is moving along a track at #12 (cm)/s#. If the curvature of the track changes from a radius of #5 cm# to #8 cm#, by how much must the centripetal force applied by the tracks change?

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