# Show that the vertical component of the reaction is constant

A smooth wire in the shape of the helix $$\vec{r} = a (\cos θ)\vec{i}+ a (\sin θ)\vec{j}+ cθ\vec{k}$$, where $$a$$ and $$c$$ are positive constants is fixed with the $$z$$-axis pointing vertically downwards. A bead of mass $$m$$, free to slide along the wire, is released from rest at the point A where $$θ = 0$$. Show that the vertical component of the reaction is constant.

I applied $$\vec{f}=m\vec{a}$$ to the particle vertically so I got

$$f=ma=R_{k}-mg=mc\ddot{\theta}$$ My textbooks solution is $$R_{k}=mc\ddot{\theta}$$

Is there anything wrong what I did?

## 1 Answer

The bead slides down a slope with a horizontal distance of 2πa and a vertical distance of 2πc. The cosine from the vertical is c/a. The acceleration down the slope is aα and the vertical component is cα. (α being the angular acceleration.) Then taking + down, mg – R = mcα. Where R (the vertical component of the reaction force) is a constant (But it is not equal to mcα).

• Thank you I mistook something – D ake Jan 5 at 20:09