Question

A 45.3-cm diameter disk rotates with a constant angular acceleration of 2.30 rad/s². It starts from rest at t = 0, and a line drawn from the center of the disk to a point P on the rim of the disk makes an angle of 57.3° with the positive x-axis at this time.

(a) Find the angular speed of the wheel at t = 2.30 s

5.26 (correct) rad/s

(b) Find the linear velocity and tangential acceleration of P at t = 2.30 s.

linear velocity- ? m/s

tangential acceleration- ? m/s²

(c) Find the position of P (in degrees, with respect to the positive x-axis) at t = 2.30s.

^? °

Answer 1

Answer :

a). We need to find the final angular speed (w) of the wheel at t= 2.30 sec.

So , final angular speed    =  initial angular speed( ) +  angular acceleration () time(t)

as it is given that the disc starts fro rest so initial angular speed will be zero .

therefore,

0 + (2.30 rad/s^2) *( 2.30 sec)

5.29 rad/sec . (Answer a)

Since the angular speed  is positive that means the disc is rotating in counterclockwise , technically speed is directionless so it doesn't really matters whether it's positive or negative .

b) to find the linear velocity and tangential acceleration of point P at t= 2.30 sec.

Linear velocity .

m/sec Linear velocity (Answer b)

tangential acceleration (at ) = radius (r) * angular acceleration ().

  (a_t ) =

  (a_t ) = ( 0.2265 meter) * 2.30 rad/s^2

  (a_t )= 0.52095 meter/sec^2 Tangential acceleration ( Answer b)

c) according to the rotational kinematics formula we find the position P in degrees with respect to the positive x-axis at t= 2.30 sec is :

(final angular speed)2 = (initial angular speed)2  + 2(angular acceleration )(angular displacement).

Which is

Since it is said that it starting from rest so

therefore ,   

(5.29 rad/sec)² / 2(2.30 rad/sec² )=

6.0835 rad =

so we convert radians to degrees ,

we get :

So , we now take this angular displacement( 348.559 degrees) and add to it 57.3 degrees

therefore , 348.559 degrees + 57.3 degrees = 405.859 degrees.

when we take co-ordinate axis and we go from  positive x-axis to complete 360 degrees angle (counterclockwise ) and we go all the way around once that is 360 degrees ( a complete round) and then we keep going until we get 405.889 degrees  . So ,

when we subtract we get :

405.859 degrees - 360 degrees = 45.859 degrees ( Answer c)

so we have to go additional 45.859 degrees to get this angle counterclockwise at positive x-axis. .