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?

The centripetal force is

The variation in centripetal force is

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.