From what minimum height of the inclined plane should the bar slide without friction in order

From what minimum height of the inclined plane should the bar slide without friction in order to describe the “dead loop” of radius R without applying pressure at the top of the loop?

R

P = 0.

hmin -?

When passing the upper point of the loop, 2 forces act on the bar: gravity Ft directed vertically downward, force N of surface pressure directed vertically downward.

m * a = m * g + N – 2 Newton’s law in vector form.

For projections onto the vertical axis 2, Newton’s law will take the form: m * a = Fт + N.

N = Fт – m * a.

The force of gravity Ft is determined by the formula: Ft = m * g.

N = m * g – m * a = m * (g – a).

The centripetal acceleration a is expressed by the formula: a = V ^ 2 / R.

N = m * (g – V ^ 2 / R).

Since at the upper point P = T = 0, then g = V ^ 2 / R.

V2 = R * g.

According to the law of conservation of energy: m * g * hmin = m * V ^ 2/2.

hmin = V ^ 2/2 * g = R * g / 2 * g = R / 2.

Answer: the minimum height is half the radius of the loop: hmin = R / 2.