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A 2.3-kg toy locomotive is pulling a 1.4-kg caboose. The frictional force of the track on the caboose is 0.46 N backward along the track. If the train is accelerating forward is 3.1 m/s2, what is the magnitude of the force exerted by the locomotive on the caboose

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The net vertical force on the caboose (C) is

∑ F[v, C] = F[n, C] - (1.4 kg) g = 0

where F[n, C] is the magnitude of the normal force exerted by the track on the caboose; so we have

F[n, C] = (2.3 kg) g = 13.72 N

which means the coefficient of kinetic friction µ between the caboose and the track is

F[f, C] = 0.46 N = µ (13.72 N) ⇒ µ = (0.46 N) / (13.72 N) ≈ 0.034

and presumably the coefficient is the same for the locomotive.

The net vertical force on the locomotive (L) is

∑ F[v, L] = F[n, L] - (2.3 kg) g = 0

where F[n, L] is the magnitude of the normal on the locomotive, so that

F[n, L] = (2.3 kg) g = 22.54 N

and so the locomotive is opposed by a frictional force with magnitude

F[f, L] = µ (22.54 N) ≈ 0.76 N

Taking the locomotive and caboose together, the net horizontal force on the system is

∑ F[h] ≈ - F[f, C] - F[f, L] + F[engine] = (2.3 kg + 1.4 kg) (3.1 m/s²)

Solve for F[engine] :

F[engine] ≈ (3.7 kg) (3.1 m/s²) + 0.46 N + 0.76 N ≈ 13 N

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