Question

Calculate (a) the torque, (b) the energy, and (c) the average power required to accelerate Earth in 4.0 days from rest to its present angular speed about its axis. Take the rotational inertia of Earth to be 9.71 x 10 37 kg m2. (Note: let one day be 24 hours)

Answer 1

Torque = 2.05 x 10²⁸ Nm

Energy = 3.54 x 10³³ J

Average power = 1.02 x 10²⁸ W

Step-by-step explanation:

(a) Torque (τ) is the rotational effect of a given force.

It is given by

τ = I x α -------------(i)

Where;

I = rotational inertia of the object

α = angular acceleration of the object.

In this case, the object is the Earth. Therefore,

I = 9.71 x 10³⁷ kg m²

α = ω / t

Where;

ω = angular velocity of earth = 2π rad / day

Since

1 day = 24 hours and 1 hour = 3600seconds

1 day = 24 x 3600 seconds = 86400seconds

=> ω = 2π rad / 86400seconds

=> ω = 7.29 × 10⁻⁵ rad/s

t = 4 days = 4 x 24 x 3600 seconds = 345600 seconds

=> α = ω / t

=> α = 7.29 × 10⁻⁵ / 345600

=> α = (7.29 × 10⁻⁵) / (3.456 x 10⁵)

=> α = (7.29 × 10⁻⁵⁻⁵) / (3.456)

=> α = (7.29 × 10⁻¹⁰) / (3.456)

=> α = 2.11 × 10⁻¹⁰ rad/s²

Now substitute the values of I and α into equation (i)

τ = 9.71 x 10³⁷ x 2.11 × 10⁻¹⁰

τ = 9.71 x 10²⁷ x 2.11

τ = 20.5 x 10²⁷ Nm

τ = 2.05 x 10²⁸ Nm

(ii) The energy (rotational energy) E is given by;

E =
`(1)/(2)` x I x ω

E =
`(1)/(2)` x 9.71 x 10³⁷ x 7.29 × 10⁻⁵

E = 35.4 x 10³² J

E = 3.54 x 10³³ J

(iii) The average power P, is given by;

P = E / t

P = 3.54 x 10³³ / 345600

P = 1.02 x 10²⁸ W