A model train with a mass of #8 kg# is moving along a track at #9 (cm)/s#. If the curvature of the track changes from a radius of #21 cm# to #120 cm#, by how much must the centripetal force applied by the tracks change?

Question

A model train with a mass of #8  kg# is moving along a track at #9 (cm)/s#.  If the curvature of the track changes from a radius of #21 cm# to #120 cm#, by how much must the centripetal force applied by the tracks change?

Sebastian Acosta · Accepted Answer

The change in centripetal force is #=0.025N#

The centripetal force is

#F=(mv^2)/r#

mass, #m=8kg#

speed, #v=0.09ms^-1#

radius, #=r#

The variation in centripetal force is

#DeltaF=F_2-F_1#

#F_1=8*0.09^2/0.21=0.31N#

#F_2=8*0.09^2/1.20=0.054N#

#DeltaF=0.31-0.054=0.025N#

Emily Abbate · Answer

undefined To determine the change in the centripetal force applied by the tracks, we first need to calculate the initial and final centripetal forces using the formula for centripetal force:

$$ F = \frac{{mv^2}}{r} $$

Where:
- $ F $ is the centripetal force,
- $ m $ is the mass of the model train,
- $ v $ is the velocity of the model train, and
- $ r $ is the radius of the curvature of the track.

Given:
- Mass of the model train, $ m = 8 $ kg
- Initial radius of curvature, $ r_1 = 21 $ cm
- Final radius of curvature, $ r_2 = 120 $ cm
- Velocity of the model train, $ v = 9 $ cm/s

We first calculate the initial centripetal force:

$$ F_1 = \frac{{mv^2}}{{r_1}} = \frac{{8 \times (9)^2}}{{21}} $$

$$ F_1 \approx \frac{{8 \times 81}}{{21}} $$

$$ F_1 \approx \frac{{648}}{{21}} $$

$$ F_1 \approx 30.86 \text{ N} $$

Then, we calculate the final centripetal force:

$$ F_2 = \frac{{mv^2}}{{r_2}} = \frac{{8 \times (9)^2}}{{120}} $$

$$ F_2 \approx \frac{{8 \times 81}}{{120}} $$

$$ F_2 \approx \frac{{648}}{{120}} $$

$$ F_2 \approx 5.4 \text{ N} $$

The change in the centripetal force applied by the tracks is the difference between the final and initial centripetal forces:

$$ \Delta F = F_2 - F_1 = 5.4 - 30.86 $$

$$ \Delta F \approx -25.46 \text{ N} $$

So, the centripetal force applied by the tracks must change by approximately $ -25.46 $ N.