Question

The rectangular object in the figure below consists of four masses connected by light rods. What power must be applied to this object to accelerate it from rest to an angular speed of 3.38 rad/s in 7.6 s for the following cases.

(a) the object is rotated about the x-axis
W

(b) the object is rotated about the y-axis
W

(c) the object is rotated about the z-axis (which is through the origin and perpendicular to the page)
W

Answer 1

change the values

The initial angular speed ?i = 0

The final angular speed ?f = 2.63rad/s

the time interval ?t = 5.2s

The work done is

                  W = ?KE = (1/2)I?²

In each situation we're accelerating from rest,

so KE_i = 0, KE_f = (1/2)I?², and ?KE = (1/2)I?²

The angular acceleration is

                ? = ?? / ?t

                   = (2.63rad/s) / (5.2 s)

                  = 0.506 rad/s²

Then power applied to the object can be given by

              P = W / ?t

                 = ?(?? / ?t)

                 = ??

And

                 ? = I?

Therefore P = (I?)(?)

The moment of inertia

               I = ?mr²

a)

Rotation about the x-axis

            I = ?mr² = ?(m_xr_x² + m_yr_y² + m_zr_z²)

              = (3.0)(0.5)^2 + (4.0)(0.5)^2 + (1.2)(0)^2 + (2.5)(0)^2

             = 1.75 kg * m²

Then power is

           P = (I?)(?)

              = (1.75 kg *m²)(0.506 rad/s²)(2.63 rad/s)

              = 2.34 W

b)

Rotation about the y-axis

            I = ?mr² = ?(m_xr_x² + m_yr_y² + m_zr_z²)

            I = (3.0 kg)(0m)² +(4.0)(0.7)² + (1.2 kg)(0.70m)² + (2.5 kg)(0m)²

              = 2.55 kg * m²

Then power

           P = (I?)(?)

              = (2.55 kg *m²)(0.506 rad/s²)(2.63 rad/s)

              = 3.39 W

c)

Rotation about the z-axis

          I = ?mr² = ?(m_xr_x² + m_yr_y² + m_zr_z²)

          I = (3.0 kg)(0.5m)² +(4.0)[(0.7)² + (0.5)² ] + (1.2 kg)(0.70m)² + (2.5 kg)(0m)²

            = 5.02 kg * m²

Then the power is

         P = (I?)(?)

            = (5.02 kg *m²)(0.506 rad/s²)(2.63 rad/s)

            = 6.68 W

Answer 2