Calculate the angular acceleration of the wheel:

A locomotive wheel of radius r = 0.8 m is illustrated in Figure. At the given instant, the speed of the locomotive is 1.6 m/sec and is accelerating at the rate of 2 m/sec². Calculate the angular acceleration of the wheel and the acceleration of the points P₁, P₂, P₃ and P₄ lying on the vertical and horizontal diameters.

Solution

We have v A  = r ω . P₂ is instantaneous centre of rotation.

∴ ω = v_A / r= 1.6/0.8 = 2 rad / sec

a _A  = r α

∴ α = a A / r =   2/0.8 = 2.5 rad / sec²

Supposing the wheel to be moving towards the right as illustrated ω and α will be in the clockwise direction.

Accelerations of P₁, P₂, etc. can be calculated very easily from the equations derived above.

a_A t^ = a α = 2 m/sec²

a _A n^ = a ω² = 3.2 m / sec ²

Point P1

tan θ =2/1.2 = 1.67

θ = 59.03^o

∴ a _p2 n = 3.2 m/sec²

Likewise for point P₂, acceleration of P₃ and P₄ can be calculated in the same way.