A railway locomotive enters a stretch of track, which is in the form o

1.	A railway locomotive enters a stretch of track, which is in the form of a circular arc of radius 280 m, at 10 m/s and with its speed increasing uniformly. Ten seconds into the stretch its speed is 14m/s and at 18s its speed is 19 m/s.Find (i) the magnitude of the locomotive’s linear acceleration when its speed is 14 m/s(ii) the direction of this acceleration at that point with respect to the locomotive’s radial acceleration (iii) the angular acceleration of the locomotive.
Answer» Data : r = 280 m, v₁ = 10 m/s at t₁ = 0, v₂ = 14 m/s at t₂= 10 s, v₃ = 19 m/s at t₃ = 18 s (i) At t = t₂ , the radial acceleration is Since the tangential acceleration is constant, it may be found from the data for any two times. (ii) If θ is the angle between the resultant linear acceleration and the radial acceleration, tan θ = \(\frac{a_t}{a_r}\) = \(\frac{0.5}{0.7}\) = 0.7142 ∴ θ = tan^-1 0.7142 = 35°32′ (iii) a_t = αr The angular acceleration, α = \(\frac{α_t}r\) = \(\frac{0.5}{280}\) = 1.785 × 10^-3 rad/s² = 1.785 mrad/s²

A railway locomotive enters a stretch of track, which is in the form of a circular arc of radius 280 m, at 10 m/s and with its speed increasing uniformly. Ten seconds into the stretch its speed is 14m/s and at 18s its speed is 19 m/s.Find (i) the magnitude of the locomotive’s linear acceleration when its speed is 14 m/s(ii) the direction of this acceleration at that point with respect to the locomotive’s radial acceleration (iii) the angular acceleration of the locomotive.

Answer»

Data : r = 280 m, v₁ = 10 m/s at t₁ = 0, v₂ = 14 m/s at t₂= 10 s, v₃ = 19 m/s at t₃ = 18 s

(i) At t = t₂ , the radial acceleration is

Since the tangential acceleration is constant, it may be found from the data for any two times.

(ii) If θ is the angle between the resultant linear acceleration and the radial acceleration,

tan θ = \(\frac{a_t}{a_r}\) = \(\frac{0.5}{0.7}\) = 0.7142

∴ θ = tan^-1 0.7142 = 35°32′

(iii) a_t = αr

The angular acceleration,

α = \(\frac{α_t}r\) = \(\frac{0.5}{280}\)

= 1.785 × 10^-3 rad/s²

= 1.785 mrad/s²