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 2.63 rad/s in 5.2 s for the following cases. Give answer in 4 significant figures

(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

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 2.63 rad/s in 5.2 s for the following cases. Give answer in 4 significant figures (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

The initial angular speed ωi = 0

 

The final angular speed ωf = 2.63rad/s

 

the time interval Δt = 5.2s

 

The work done is

 

                  W = ΔKE = (1/2)Iω²

 

In each situation we're accelerating from rest,

 

so KE_i = 0, KE_f = (1/2)Iω², and ΔKE = (1/2)Iω²

 

The angular acceleration is 

 

                α = Δω / Δt

 

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

 

                  = 0.506 rad/s²

Then power applied to the object can be given by 

 

              P = W / Δt

 

                 = τ(Δθ / Δt)

 

                 = τω

 

And 

 

                 τ = Iα 

 

Therefore P = (Iα)(ω)

 

 

 

The moment of inertia

               I = Σmr²

 

a)

 

Rotation about the x-axis 

 

            I = Σmr² = Σ(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 = (Iα)(ω)

 

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

 

              = 2.34 W 

 

b)

Rotation about the y-axis

            I = Σmr² = Σ(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 = (Iα)(ω)

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

              = 3.39 W

c)

Rotation about the z-axis 

          I = Σmr² = Σ(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 = (Iα)(ω)

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

            = 6.68 W

 

 

Answer 2

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