A model train with a mass of #4 kg# is moving along a track at #21 (cm)/s#. If the curvature of the track changes from a radius of #84 cm# to #35 cm#, by how much must the centripetal force applied by the tracks change?

Question

A model train with a mass of #4  kg# is moving along a track at #21 (cm)/s#.  If the curvature of the track changes from a radius of #84 cm# to #35 cm#, by how much must the centripetal force applied by the tracks change?

Charlotte Aaron · Accepted Answer

The change in centripetal force is #=0.29N#

The centripetal force is

#F=(mv^2)/r#

mass, #m=4kg#

speed, #v=0.21ms^-1#

radius, #=(r) m#

The variation in centripetal force is

#DeltaF=F_2-F_1#

#F_1=mv^2/r_1=4*0.21^2/0.84=0.21N#

#F_2=mv^2/r_2=4*0.21^2/0.35=0.504N#

#DeltaF=0.504-0.21=0.29N#

Lily Adams · Answer

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

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

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

First, we calculate the initial centripetal force when the radius is 84 cm:

$$ F_{c1} = \frac{(4 \, \text{kg}) \times (0.21 \, \text{m/s})^2}{0.84 \, \text{m}} $$

$$ F_{c1} = \frac{(4 \, \text{kg}) \times (0.0441 \, \text{m/s}^2)}{0.84 \, \text{m}} $$

$$ F_{c1} = \frac{0.1764 \, \text{N}}{0.84 \, \text{m}} $$

$$ F_{c1} ≈ 0.21 \, \text{N} $$

Next, we calculate the final centripetal force when the radius is 35 cm:

$$ F_{c2} = \frac{(4 \, \text{kg}) \times (0.21 \, \text{m/s})^2}{0.35 \, \text{m}} $$

$$ F_{c2} = \frac{(4 \, \text{kg}) \times (0.0441 \, \text{m/s}^2)}{0.35 \, \text{m}} $$

$$ F_{c2} = \frac{0.1764 \, \text{N}}{0.35 \, \text{m}} $$

$$ F_{c2} ≈ 0.504 \, \text{N} $$

Finally, we find the change in centripetal force:

$$ \Delta F_c = F_{c2} - F_{c1} $$

$$ \Delta F_c = 0.504 \, \text{N} - 0.21 \, \text{N} $$

$$ \Delta F_c ≈ 0.294 \, \text{N} $$

So, the centripetal force applied by the tracks must increase by approximately $ 0.294 \, \text{N} $ when the curvature of the track changes from a radius of 84 cm to 35 cm.