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archaic

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- Homework Statement
- The apparatus in the figure is a distance ##h## from the ground. The cylinder of radius ##r## rolls without slipping. I am asked to find the total horizontal distance that will be travelled until the rod reaches the ground.

- Relevant Equations
- $$\mathbf{F_B}=I\mathbf L\times\mathbf B$$$$\tau_{net}=I\alpha$$

Hello!

The magnetic force is to the right. ##I_c## is the moment of inertia of the cylinder.

For the net force on the centre of mass, I have the frictional and magnetic forces ##F=F_B-f##. I know that ##F_B## is ##IdB##.

I also know that ##rf=I_c\alpha=I_c\frac ar##, so that ##f=I_c\frac{a}{r^2}=\frac{ma}{2}## (because ##I_c=\frac12mr^2##), which gives me ##ma=IdB-\frac{ma}{2}\Leftrightarrow a=\frac{2IdB}{3m}\Leftrightarrow F=\frac{2IdB}{3}##.

Using the work energy theorem, I get ##\frac12mv^2=\frac23IdBL\Leftrightarrow v=\sqrt{\frac{4IdBL}{3m}}##, which is the horizontal speed when the rod enters free-fall.

The vertical position when the cylinder touches the ground is such that ##0=h-\frac12g(\Delta t)^2\Leftrightarrow\Delta t=\sqrt{\frac{2h}{g}}##.

The total range would be ##L+v\Delta t=L+\sqrt{\frac{8IdBLh}{3mg}}##

Correct?

Thank you. :)

The magnetic force is to the right. ##I_c## is the moment of inertia of the cylinder.

For the net force on the centre of mass, I have the frictional and magnetic forces ##F=F_B-f##. I know that ##F_B## is ##IdB##.

I also know that ##rf=I_c\alpha=I_c\frac ar##, so that ##f=I_c\frac{a}{r^2}=\frac{ma}{2}## (because ##I_c=\frac12mr^2##), which gives me ##ma=IdB-\frac{ma}{2}\Leftrightarrow a=\frac{2IdB}{3m}\Leftrightarrow F=\frac{2IdB}{3}##.

Using the work energy theorem, I get ##\frac12mv^2=\frac23IdBL\Leftrightarrow v=\sqrt{\frac{4IdBL}{3m}}##, which is the horizontal speed when the rod enters free-fall.

The vertical position when the cylinder touches the ground is such that ##0=h-\frac12g(\Delta t)^2\Leftrightarrow\Delta t=\sqrt{\frac{2h}{g}}##.

The total range would be ##L+v\Delta t=L+\sqrt{\frac{8IdBLh}{3mg}}##

Correct?

Thank you. :)

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