Rotational Work and Energy

The rotational work done by a force about the fixed axis of rotation is defined as

Wrot =   (10.23)

Where τ is the torque produced by the force, and dθ is the infinitesimally small angular displacement about the axis.

The rotational kinetic energy of a body about a fixed rotational axis is defined as

Krot = 1/2 Iω₂ (10.24)

where I is the moment of inertia about the axis.

Work – Energy Theorem

In complete analog to the work energy theorem for the translatory motion, it can be stated for rotational motion as:

Wrot = ΔKr_ot (10.25)

The net rotational work done by the forces is equal to the change in rotational kinetic energy of the body.

Conservation of Mechanical Energy

In the absence of dissipative work done by non-conservative forces, the total mechanical energy of a system is conserved.

ΔK + ΔU = 0

or Kf + Uf = Ki + Ui (10.26)

Example 10.12

When the block falls by a distance x, its potential energy decreases (ΔUg = -mgx); the potential energy of the spring increases (ΔUs = +1/2kx2), and both the block and the pulley gain kinetic energy (ΔK = 1/2mv2 + 1/2Iω₂).

From the conservation of mechanical energy, ΔK + ΔU = 0,

Note that R was not needed.

Putting m = 4 kg; M = 8 kg; k = 32 N/m; x = 1m

4v₂ + 16 – 40 = 0

orv = 2.4 m/s

Rotational Power

In complete analog with the linear motion, the instantaneous rotational power is defined as

Prot = (10.27)

Therefore, τ = TR = (500)(0.25) = 125 Nm.

Angular velocity, ω = 2πN/60

or ω = 2π(20)/60 = 2π/3 rad/s

The power required is

P = τω = (125 N m) = 2π/3 rad/s = 260 W