To determine the revolution of a pulley in 2 seconds given the acceleration and initial velocity of the cable, we need to use kinematic equations.
Let's assume that the pulley starts from rest and rotates in a counterclockwise direction. The linear motion of the cable can be related to the angular motion of the pulley using the following equations:
The equation relating angular displacement (θ), angular velocity (ω), and time (t) is given by: θ = ω₀t + (1/2)αt² where ω₀ is the initial angular velocity, α is the angular acceleration, and t is the time.
The equation relating angular velocity, angular acceleration, and time is given by: ω = ω₀ + αt
In this case, the angular acceleration can be determined from the linear acceleration of the cable since the pulley radius is constant. The linear acceleration (a) is related to the angular acceleration (α) by: a = αr where r is the radius of the pulley.
Given that the linear acceleration of the cable (a) is 9 inch/sec² and the initial velocity (v₀) is 12 inch/sec, we can convert these values to angular acceleration (α) and initial angular velocity (ω₀) using the relationship: α = a/r ω₀ = v₀/r
Assuming the radius of the pulley is known, let's proceed with the calculation.