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 #88 cm# to #28 cm#, by how much must the centripetal force applied by the tracks change?

Centripetal force is what

The centripetal force fluctuation is

To calculate the change in centripetal force, we first need to determine the initial and final centripetal forces acting on the train as it moves along the track. The centripetal force is given by the equation:

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

where:

• ( m ) = mass of the train (5 kg)
• ( v ) = velocity of the train (14 cm/s)
• ( r ) = radius of curvature of the track

Using the initial radius of curvature (( r = 88 ) cm), we can calculate the initial centripetal force (( F_{c1} )):

[ F_{c1} = \frac{(5 , \text{kg}) \times (14 , \text{cm/s})^2}{88 , \text{cm}} ]

Then, using the final radius of curvature (( r = 28 ) cm), we can calculate the final centripetal force (( F_{c2} )):

[ F_{c2} = \frac{(5 , \text{kg}) \times (14 , \text{cm/s})^2}{28 , \text{cm}} ]

Finally, we can find the change in centripetal force (( \Delta F_c )) by subtracting the initial force from the final force:

[ \Delta F_c = F_{c2} - F_{c1} ]

Plugging in the values and calculating:

[ F_{c1} = \frac{(5 , \text{kg}) \times (14 , \text{cm/s})^2}{88 , \text{cm}} = 10 , \text{N} ] [ F_{c2} = \frac{(5 , \text{kg}) \times (14 , \text{cm/s})^2}{28 , \text{cm}} = 50 , \text{N} ]

[ \Delta F_c = F_{c2} - F_{c1} = 50 , \text{N} - 10 , \text{N} = 40 , \text{N} ]

So, the centripetal force applied by the tracks must increase by 40 N when the curvature of the track changes from a radius of 88 cm to 28 cm.