A sheet of material, mass 2 kg, uniform density, is in the shape of a right tria

Question

A sheet of material, mass 2 kg, uniform density, is in the shape of a right triangle bounded by the lines y=0, x=4, and y=x. It is attached to the origin, and free to rotate about the y axis. A force of 1000 newtons is applied perpendicular to the plane, located at the x=4 edge, and continues to be applied constantly to the same edge as it starts to rotate. In a right-handed coordinate system, how fast will the object be spinning as it crosses the y-z plane (that is, when it has rotated through 270 degrees)?

Explanation / Answer

The moment of inertia about an axis passing through origin perpendicular to the plane

is I = 2m[(h2/12) + (b2/4)]

    I = 2*2[(42/12) + (42/4)] =(64/3) kgm2

When it rotates through: ? = 270 deg = (3?/2) rad by applying a force of F = 1000 N

Torque acting on this sheet is,

      ? = F*perpendicular distance

Here perpendicular distance = 4 m

Thus, ? =1000*4 = 4000 Nm

Torque is also defined as, ? = I?

Thus, 4000=  (64/3) *?

        ? =187.5 rad/s2

Apply rotational kinematics:

  ?f2 -?i2 =2??

?i = initial angular velocity = 0

?f = final angular speed

?f2 = 2*187.5*3?/2

Therefore, the speed of rotation of the object is,

         ?f = 42.02 rad/sec

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