Question

A disk 8.04 cm in radius rotates at a constant rate of 1 220 rev/min about its central axis. (a) Determine its angular speed. rad/s (b) Determine the tangential speed at a point 3.02 cm from its center. m/s (c) Determine the radial acceleration of a point on the rim. magnitude km/s2 direction (d) Determine the total distance a point on the rim moves in 2.02 s. m

Answer 1

A) 128 rad/s

B) 3.87 m/s

C) 1317.17 m/s²

D) 20.79m

Step-by-step explanation:

A) We are given the angular speed as 1220 rev/min. Now let's convert it to rad/s.

1220 (rev/min) x (2πrad/1rev) x (1min/60sec) = (1220 x 2π)/60 rad/s = 127.76 rad/s ≈ 128 rad/s

B) Formula for tangential speed is given as;

v = ωr

We know that ω = 128 rad/s

Also, r = 3.02cm = 0.0302m

Thus, v = 128 x 0.0302 = 3.87 m/s

C) Formula for radial acceleration is given as;

a_c = v²/r

From earlier, v = ωr

Thus, a_c = v²/r = (ωr)²/r = ω²r

On the rim, r = 8.04cm = 0.0804

a_c = 128² x 0.0804 = 1317.17 m/s²

D) We know that; distance/time = speed

Thus, distance = speed x time

D = vt

From earlier, v = ωr

Thus, D = ωrt

Plugging in the relevant values ;

D = 128 x 0.0804 x 2.02 = 20.79m

Answer 2

a) 23.04 rad/s

b) 0.79 m/s

c) 16 m/s²

d) 1.41 m

Step-by-step explanation:

given,

radius of disk, r = 8.04 cm = 0.0804 m

Speed of disk, = 220 rpm

a) to get the angular speed in rad/s is quite easy since we already have it in rpm. All that is needed is conversion from rpm to rad/s

ω rad/s = 220 rpm * 2π rad/rev * 1/60 min/s

ω rad/s = 1382.48 / 60

ω rad/s = 23.04 rad/s

b) tangential speed at a point, 3.02 cm (0.0302 m) from the centre

Tangential speed is usually gotten from angular speed. Such that,

v = ωr

v = 23.04 * 0.0302

v = 0.70 m/s

c) radial acceleration of a point on the rim can also be expressed in terms of angular speed, such that.

a(r) = ω²r

a(r) = 23.04² * 0.0302

a(r) = 530.84 * 0.0302

a(r) = 16 m/s² directed towards the centre

d) total distance at a point on the rim.

The rim moves in 2.02 s, so,

d = ωrt

d = 23.04 * 0.0302 * 2.02

d = 1.41 m