1. a flywheel is making 180 r.p.m and after 20 second it is running at 140 r.p.m. how many revolutions will it make, and what time will elapse before it stop, if the retardation is uniform?
See Related Questions: Wed, 12-31-14, 5:48 AM.
Answer
To find the number of revolutions the flywheel will make and the time it will take to stop, we need to use the concept of uniformly retarded motion. Uniform retardation means that the rate of change of angular velocity is constant.
Given information:
Initial angular velocity, ω₁ = 180 r.p.m.
Final angular velocity, ω₂ = 140 r.p.m.
Time, t = 20 seconds
First, let's convert the angular velocities from revolutions per minute (r.p.m.) to radians per second (rad/s). Since 1 revolution is equal to 2π radians:
ω₁ = 180 r.p.m. = 180 * 2π / 60 rad/s = 6π rad/s
ω₂ = 140 r.p.m. = 140 * 2π / 60 rad/s = (7π/3) rad/s
Now, we can apply the formula for uniformly retarded motion:
ω₂ = ω₁ + αt
where α is the angular acceleration and t is the time.
Rearranging the formula to solve for α:
α = (ω₂ - ω₁) / t
Substituting the given values:
α = ((7π/3) - 6π) / 20 = (-5π/3) / 20 = -π/12 rad/s²
Now, we can use the formula for angular displacement with uniform retardation:
θ = ω₁t + (1/2)αt²
Since we want to find the number of revolutions, we need to convert the displacement from radians to revolutions:
θ = (ω₁t + (1/2)αt²) / 2π
Substituting the values:
θ = (6π * 20 + (1/2)(-π/12)(20²)) / (2π) = (120π - (50π/12)) / (2π) = (120 - 50/12) / 2 = 110/12 = 9.17 revolutions
So, the flywheel will make approximately 9.17 revolutions.
To find the time it takes to stop, we can use the formula:
ω₂ = ω₁ + αt
Solving for t:
t = (ω₂ - ω₁) / α
Substituting the values:
t = (7π/3 - 6π) / (-π/12) = π / (π/12) = 12 seconds
Therefore, it will take 12 seconds for the flywheel to stop.