A model train, with a mass of #32 kg#, is moving on a circular track with a radius of #9 m#. If the train's kinetic energy changes from #18 j# to #21 j#, by how much will the centripetal force applied by the tracks change by?

Question

A model train, with a mass of #32 kg#, is moving on a circular track with a radius of #9 m#.  If the train's kinetic energy changes from #18 j# to #21 j#, by how much will the centripetal force applied by the tracks change by?

Leo Abeyta · Accepted Answer

The change in centripetal force is #=2/3N#

The centripetal force is

#F=(mv^2)/r#

The kinetic energy is

#KE=1/2mv^2#

The variation of kinetic energy is

#Delta KE=1/2mv^2-1/2m u^2#

#=1/2m(v^2-u^2)#

The mass is #m=32kg#

The radius of the track is #r=9m#

The variation of centripetal force is

#DeltaF=m/r(v^2-u^2)#

#DeltaF=2m/r1/2(v^2-u^2)#

#=(2)/r*1/2m(v^2-u^2)#

#=(2)/r*Delta KE#

#=2/9*(21-18)N#

#=2/3N#

Adrian Ayala · Answer

undefined The centripetal force applied by the tracks can be calculated using the formula:

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

where $ m $ is the mass of the train, $ v $ is the velocity of the train, and $ r $ is the radius of the circular track.

Given that the mass of the train is $ 32 \, \text{kg} $ and the radius of the track is $ 9 \, \text{m} $, we can calculate the initial centripetal force ($ F_1 $) when the kinetic energy is $ 18 \, \text{J} $:

$$ F_1 = \frac{m \cdot (v_1)^2}{r} $$

where $ v_1 $ is the initial velocity of the train.

Given that the initial kinetic energy ($ E_1 $) is $ 18 \, \text{J} $, we can use the formula for kinetic energy to find $ v_1 $:

$$ E_1 = \frac{1}{2} m \cdot (v_1)^2 $$

Solving for $ v_1 $:

$$ v_1 = \sqrt{\frac{2 \cdot E_1}{m}} $$

Given that the final kinetic energy ($ E_2 $) is $ 21 \, \text{J} $, we can find the final velocity ($ v_2 $) using the formula for kinetic energy:

$$ E_2 = \frac{1}{2} m \cdot (v_2)^2 $$

Solving for $ v_2 $:

$$ v_2 = \sqrt{\frac{2 \cdot E_2}{m}} $$

Once we have $ v_1 $ and $ v_2 $, we can calculate the initial and final centripetal forces ($ F_1 $ and $ F_2 $) using the formula:

$$ F = \frac{m \cdot v^2}{r} $$

Then, the change in centripetal force is given by:

$$ \Delta F = F_2 - F_1 $$