Question

In a washing machine, the inside (cylindrical basin) rotates in a circular motion. While its spinning, it rotates at a constant angular speed at 500 rpm. While it is spinning like this, all the clothes are flung around the walls inside and thus ends up experiencing a centripetal acceleration that is 54 times the acceleration of Earth's gravity.

1) Calculate the radius of this cylindrical basin.

2) Calculate (in mph) the clothes tangential speed.

Now let's say the washing machine starts at rest and speeds up past 500 rpm while having a constant angular acceleration. At the moment when it reaches 500 rpm, the wet clothes that are flung around the walls experience a total acceleration that is equal to 54.1 times the acceleration of Earth's gravity.

3) Calculate the tangential acceleration of these clothes during this exact moment.

4) Calculate angular acceleration of the machine as it speeds up (in rad/s^2)

5) Calculate how many rotations did the machine make from rest to its angular speed of 400 rpm?

Answer 1

Frequency of rotation, f = 500 rpm = 500/60 rps = 50/6 rps

Angular frequency, ω = 2πf = 2x 3.14 x 50/6 = 52.33 rad/s

1.given:

centripetal acceleration that is 54 times the acceleration of Earth's gravity

ac = 54 g = 54 x 9.8 = 529.2 m/s2

but, centripetal acceleration, ac = r ω2

               539.2 = r x 52.33 x 52.33

Radius, r = 0.19 m

the radius of this cylindrical basin, r = 0.19 m

2.

Tangential speed, V = rω = 0.19 x 52.33 = 9.94 m/s

3. total acceleration, a = 54.1 x g = 54.1 x 9.8 = 530.18 m/ s2

But, a2 = ac2 - at2

at2 = a2 - ac2 = 530.182 – 529.22 = 1038.8

Tangential acceleration, at = 32.2 m/ s2

4.

Angular acceleration, α = a/r = 32.2/0.19 = 169.5 rad/s2

5.

Initial angular speed, ω1 = 0

Final angular speed, ω2 = 2πf = 2x 3.14 x 400/60= 41.86 rad/s

We know that,

ω22 = 2αθ

θ = ω22 / 2α = 41.86x41.86/ 2x 169.5 = 5.2 rad

if n is the number of rotations, then θ = 2πn

n= θ / 2π = 5.2/ 2 x 3.14 = 0.82

number of rotations, n = 0.82