A model train, with a mass of #3 kg#, is moving on a circular track with a radius of #8 m#. If the train's rate of revolution changes from #1/4 Hz# to #1/7 Hz#, by how much will the centripetal force applied by the tracks change by?

Question

A model train, with a mass of #3 kg#, is moving on a circular track with a radius of #8 m#.  If the train's rate of revolution changes from #1/4 Hz# to #1/7 Hz#, by how much will the centripetal force applied by the tracks change by?

Leo Abeyta · Accepted Answer

The change in centripetal force is #=39.88N#

Centripetal force is what

#F=(mv^2)/r=mromega^2N#

The mass of the train, #m=(3)kg#

The radius of the track, #r=(8)m#

The frequencies that are

#f_1=(1/4)Hz#

#f_2=(1/7)Hz#

The angular velocity is #omega=2pif#

The centripetal force fluctuation is

#DeltaF=F_2-F_1#

#F_1=mromega_1^2=mr*(2pif_1)^2=3*8*(2pi*1/4)^2=59.22N#

#F_2=mromega_2^2=mr*(2pif_2)^2=3*8*(2pi*1/7)^2=19.34N#

#DeltaF=|F_2-F_1|=59.22-19.34=39.88N#

Amelia Adams · Answer

undefined To find the change in centripetal force, we first calculate the initial and final centripetal forces using the formulas:

Initial frequency: $ f_1 = \frac{1}{4} $ Hz
Initial angular velocity: $ \omega_1 = 2\pi f_1 $
Initial centripetal force: $ F_1 = m \cdot r \cdot \omega_1^2 $

Final frequency: $ f_2 = \frac{1}{7} $ Hz
Final angular velocity: $ \omega_2 = 2\pi f_2 $
Final centripetal force: $ F_2 = m \cdot r \cdot \omega_2^2 $

Substitute the values and find the change in centripetal force:

$$ \Delta F = F_2 - F_1 $$

$$ \Delta F = m \cdot r \cdot \omega_2^2 - m \cdot r \cdot \omega_1^2 $$

$$ \Delta F = m \cdot r \cdot ( \omega_2^2 - \omega_1^2 ) $$

$$ \Delta F = m \cdot r \cdot ( (2\pi f_2)^2 - (2\pi f_1)^2 ) $$

$$ \Delta F = m \cdot r \cdot ( (2\pi \cdot \frac{1}{7})^2 - (2\pi \cdot \frac{1}{4})^2 ) $$

Calculate the values and subtract to find the change in centripetal force.