What is the new angular speed of the merry-go-round after a child hops on?

[Paymemoney]

Homework Statement

A playground merry-go-round of radius R=2.00m has a moment of intertia $$I=250kg.m^2$$ and is rotating at 10 rev/min (rpm) about a frictionless vertical axle. Facing the axle, a 25.0kg child hops onto the merry-go-round from the ground and manages to sit down on its edge. What is the new angular speed of the merry-go-round?

Homework Equations

$$\tau=rF$$

$$\tau=I\alpha$$

$$\omega final = \omega initial + \alpha t$$

The Attempt at a Solution

I have done this, however my answer is incorrect.

$$\tau=rF$$

$$\tau=2*25a$$

$$\tau=50a$$

$$\tau=Im^2$$

$$50a=250\alpha$$

$$v=\frac{x}{t}$$

$$1.047=\frac{2}{t}$$

$$t=2.094$$

so...
$$50*{\omega}{t}=250\alpha$$

$$50*\frac{1.047}{2.094}=250\alpha$$

$$\alpha=\frac{25}{250}$$

$$\alpha=0.1$$

now to find the final angular speed
$$\omega final = \omega initial + \alpha t$$

$$\omega final = 1.047 + 0.1*2.094$$

$$\omega final = 1.2564rad/s$$

Also a quick question can someone explain to me what is the difference between the
$$\tau=rF$$ & $$\tau=I\alpha$$ equations??
I'm not quite sure if this is correct; Is the $$\tau=rF$$ normally used for small rotating mass, and $$\tau=I\alpha$$ is used for a large rotating body.

P.S

 

[tiny-tim]

Hi Paymemoney!

(have a tau: τ and an omega: ω and an alpha: α and try using the X² tag just above the Reply box )

A playground merry-go-round of radius R=2.00m has a moment of intertia $$I=250kg.m^2$$ and is rotating at 10 rev/min (rpm) about a frictionless vertical axle. Facing the axle, a 25.0kg child hops onto the merry-go-round from the ground and manages to sit down on its edge. What is the new angular speed of the merry-go-round?

Homework Equations

$$\tau=rF$$ …

No … "facing the axle" mans that there is no external torque …

start again, using only conservation of angular momentum. ​

Also a quick question can someone explain to me what is the difference between the
$$\tau=rF$$ & $$\tau=I\alpha$$ equations??
I'm not quite sure if this is correct; Is the $$\tau=rF$$ normally used for small rotating mass, and $$\tau=I\alpha$$ is used for a large rotating body.

τ = r x F tells you how much the torque is

τ = Iα tells you what the torque does

(and so r x F = Iα tells you what the force does)

 

[Paymemoney]

When you are talking about what the torque does, do you mean if it slowing down or accelerating?

 

[tiny-tim]

more than that …

i mean that, just as (linear) force F tells you the (linear) acceleration a (of a body with mass m),

torque τ tells you the angular acceleration α (of a body with moment of inertia I)

… they both tell you exactly what the force or torque does to the body ​