The blades of a fan running at low speed turn at 200 rpm. When the fan is switched to high speed, the rotation rate increases uniformly to 340 rpm in 5.92 s.
How many revolutions do the blades go through while the fan is accelerating?
change rpm to rev per second
wf^2=wi^2+2*a*revolutions if wf,wi are in rev/sec, a is rev/sec^2
a= wfinalinrev/sec / 5.92sec
The magnetic field in a cyclotron is 0.50T. What is the magnitude of the magnetic force on a proton with velocity of 1.0 x 107 m/s in a plane perpendicular to the field?
To find the number of revolutions the blades go through while the fan is accelerating, we first need to find the angular acceleration (α) of the blades.
The angular acceleration can be found using the formula:
α = Δω / Δt
where:
Δω is the change in angular velocity (from low speed to high speed), and
Δt is the time taken for the change in angular velocity.
In this case, the change in angular velocity is given by:
Δω = 340 rpm - 200 rpm = 140 rpm
Converting the units of angular velocity from rpm to radians per second:
Δω = 140 rpm * (2π radians / 1 min) * (1 min / 60 s) = 14π rad/s
The time taken for the change in angular velocity is given as 5.92 seconds:
Δt = 5.92 s
Now, we can find the angular acceleration:
α = Δω / Δt = 14π rad/s / 5.92 s ≈ 7.47 rad/s²
The number of revolutions the blades go through while the fan is accelerating can be calculated using the equation:
θ = ω_i * t + 0.5 * α * t²
where:
θ is the angle of rotation,
ω_i is the initial angular velocity (in radians per second),
t is the time taken (in seconds), and
α is the angular acceleration.
In this case, the initial angular velocity is given as 200 rpm:
ω_i = 200 rpm * (2π radians / 1 min) * (1 min / 60 s) = 20π rad/s
The time taken for the change in angular velocity is 5.92 seconds:
t = 5.92 s
Plugging in the values:
θ = 20π rad/s * 5.92 s + 0.5 * 7.47 rad/s² * (5.92 s)²
Calculating this expression will give us the number of revolutions the blades go through while the fan is accelerating.