Vaidity of rotational formulae for variable acceleration

[spaghetti3451]

Homework Statement

Which of the following formulas is valid if the angular acceleration of an object is not constant? Explain your reasoning in each case.

(a) v = rω;
(b) a_tan = r[itex]\alpha[/itex];
(c) ω = ω₀ + [itex]\alpha[/itex]t;
(d) a_rad = rω²;
(e) K = 0.5Iω².

Homework Equations

The Attempt at a Solution

(a) v = rω is derived from s = r[itex]\theta[/itex], where s is the arc length of the circular path, r is the radius and [itex]\theta[/itex] is the angular displacement. s = r[itex]\theta[/itex] is valid for each instant of time, whether the angular accleration is constant or not, so v = rω is valid for variable angular acceleration.

(b) Don't know!

(c) Not valid as the assumption of constant accleration leads to the formula.

(d) Don't know!

(e) The formula follows from the combination of K = 0.5mv² and v = rω. v = rω is valid for any acceleration, so K = 0.5Iω² is valid as well.

Any comments would be wel appreciated!

 

[Simon Bridge]

You are saying that a and e work for a particular instant and c is a kinematic equation...

ω(t) ≠ ω₀ + α(t)t ... would be correct.

For b and d you have presumably rejected the kind of argument you used at a. Why is that.

 

[spaghetti3451]

(b) a_tan = [itex]\frac{dv}{dt}[/itex] = [itex]\frac{d(rω)}{dt}[/itex] = r[itex]\alpha[/itex]. In other words, the tangential component of the linear acceleration of a particle within a rigid body is the rate of change of the linear speed of the particle. The linear speed depends on the angular speed of the body. In the formula, the angular speed is assumed to be variable (i.e. a function of time). Moreover, it is not constrained to be linear in time. This implies that the angular acceleration is a function of time. So, (b) is valid for variable acceleration.

(d) a_rad = [itex]\frac{v²}{r}[/itex] = ω²r. In other words, the radial component of the linear acceleration of a particle within a rigid body depends on the linear speed of the particle. The linear speed is a function of the angular speed of the particle. The angular speed is not constrained to be linear in time. This implies that the angular acceleration is a function of time. So, (d) is valid for variable acceleration.

(a) and (e) also do not constrain the angular speed to be linear in time. So, the angular acceleration is a function of time, i.e. those formulae are valid for variable acceleration.

I think this argument is much better than my previous one, because although (a) and (e) are valid for each instant of time, that doesn't hint on any constraint on ω.

Any comments would be greatly appreciated.