A model train with a mass of #2 kg# is moving along a track at #18 (cm)/s#. If the curvature of the track changes from a radius of #9 cm# to #3 cm#, by how much must the centripetal force applied by the tracks change?

Centripetal force is what

The centripetal force fluctuation is

To determine how much the centripetal force applied by the tracks must change, we need to calculate the centripetal force before and after the curvature of the track changes.

The centripetal force acting on the train as it moves along a curved track is given by the formula:

[ F_{\text{centripetal}} = \frac{mv^2}{r} ]

Where:

• ( m ) is the mass of the train (2 kg)
• ( v ) is the velocity of the train (18 cm/s)
• ( r ) is the radius of the curvature of the track (initially 9 cm, then changes to 3 cm)

First, we calculate the centripetal force before the curvature changes:

[ F_{\text{initial}} = \frac{2 \times (18)^2}{9} ]

Next, we calculate the centripetal force after the curvature changes:

[ F_{\text{final}} = \frac{2 \times (18)^2}{3} ]

Finally, we find the difference in centripetal force:

[ \Delta F = F_{\text{final}} - F_{\text{initial}} ]

Substituting the values and calculating:

[ \Delta F = \frac{2 \times (18)^2}{3} - \frac{2 \times (18)^2}{9} ]

[ \Delta F = \frac{2 \times 18^2}{3} \left(1 - \frac{1}{3}\right) ]

[ \Delta F = \frac{2 \times 18^2}{3} \times \frac{2}{3} ]

[ \Delta F = \frac{4 \times 18^2}{9} ]

[ \Delta F = \frac{4 \times 324}{9} ]

[ \Delta F = \frac{1296}{9} ]

[ \Delta F = 144 ]

So, the centripetal force applied by the tracks must change by ( 144 , \text{N} ) when the curvature of the track changes from a radius of 9 cm to 3 cm.