A model train with a mass of $4 k g$ $4 kg$ is moving along a track at $18 \frac{c m}{s}$ $18 (cm)/s$ . If the curvature of the track changes from a radius of $25 c m$ $25 cm$ to $42 c m$ $42 cm$ , by how much must the centripetal force applied by the tracks change?

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A model train with a mass of $4 k g$ $4 kg$ is moving along a track at $18 \frac{c m}{s}$ $18 (cm)/s$ . If the curvature of the track changes from a radius of $25 c m$ $25 cm$ to $42 c m$ $42 cm$ , by how much must the centripetal force applied by the tracks change?

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Answer 1 · 2016-10-20T11:24:24

The centripetal force changes in a factor of $\frac{25}{42}$ $25/42$ , i.e. approximately $0.6$ $0.6$ times greater.

Explanation:

The centripetal force acting on a moving mass $m$ $m$ traveling a circular path with radius $r$ $r$ at a constant speed $v$ $v$ is given by the formula:

$F_{c} = m \frac{v^{2}}{r}$ $F_c = m v^2/r$

If the path's radius is modified from a $r_{1}$ $r_1$ value to a $r_{2}$ $r_2$ one, the initial centripetal force $F_{c 1}$ $F_{c1}$ changes to a new value $F_{c 2}$ $F_{c2}$ which can be compared using the above formula:

$\frac{F_{c 2}}{F_{c 1}} = \frac{m \frac{v^{2}}{r_{2}}}{m \frac{v^{2}}{r_{1}}} = \frac{\frac{1}{r_{2}}}{\frac{1}{r_{1}}} = \frac{r_{1}}{r_{2}}$ ${F_{c2}}/{F_{c1}} = {m v^2/r_2}/{m v^2/r_1} = {1/r_2}/{1/r_1}=r_1 / r_2$

Thus:

${F_{c2}}/{F_{c1}} = r_1 / r_2 = {25 cm} / {42 cm} ~~0.595... rArr F_{c2} ~~0.595 F_{c1}$

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A model train with a mass of $4 k g$ $4 kg$ is moving along a track at $18 \frac{c m}{s}$ $18 (cm)/s$ . If the curvature of the track changes from a radius of $25 c m$ $25 cm$ to $42 c m$ $42 cm$ , by how much must the centripetal force applied by the tracks change?

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Explanation:

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A model train with a mass of 4 kg4kg4 kg is moving along a track at 18 (cm)/s18cms18 (cm)/s. If the curvature of the track changes from a radius of 25 cm25cm25 cm to 42 cm42cm42 cm, by how much must the centripetal force applied by the tracks change?

1 Answer

Explanation:

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A model train with a mass of $4 k g$ $4 kg$ is moving along a track at $18 \frac{c m}{s}$ $18 (cm)/s$ . If the curvature of the track changes from a radius of $25 c m$ $25 cm$ to $42 c m$ $42 cm$ , by how much must the centripetal force applied by the tracks change?