Question

Небольшая шайба массой m= 0,2 после толчка приобретает скорость v= 3 м/с И скользит по внутренней поверхности гладкого закреплённого кольца радиусом R= 0,14 м . С какой силой F шайба давит на поверхность кольца в тот момент , когда она находится на высоте h= 0,2 м от нижней точки кольца?

Answer 1

M = 0,2 г
v₀ = 3 м/с
R = 0,14 м
h = 0,2 м
F - ?
F + m*gy = m*v²/R
m*v₀²/2 = m*g*h + m*v²/2 => v² = v₀² - 2*g*h
m*gy = m*g * cos α
cos α = (h - R)/R => m*gy = m*g*(h - R)/R
F + m*g*(h - R)/R = m*(v₀² - 2*g*h)/R
F = m*(v₀² - 2*g*h)/R - m*g*(h - R)/R
F = m*(v₀² - 2*g*h - g*(h - R))/R = m*(v₀² - 2*g*h -g*h +g*R)/R = m*(v₀² - 3*g*h +g*R) / R
F = 0,2*10⁻³ кг * ((3 м/с)² - 3 * 10 м/с² * 0,2 м + 10 м/с² * 0,14 м) / 0,14 м = 0,2*10⁻³ кг * (9 м²/с² - 6 м²/с² + 1,4 м²/с²) / 0,14 м = 0,2*10⁻³  кг * 4,4 м²/с² / 0,14 м ≈ 6,3*10⁻³ Н

Answer 2

Problem Analysis

A small disc with a mass of m = 0.2 kg is given an initial velocity of v = 3 m/s and slides along the inner surface of a smooth, fixed ring with a radius of R = 0.14 m. The disc is at a height of h = 0.2 m from the bottom of the ring. We need to calculate the force F exerted by the disc on the surface of the ring at this moment.

Solution

To solve for the force F, we can use the principles of centripetal force and gravitational force. The centripetal force is the force required to keep an object moving in a circular path, and it acts perpendicular to the velocity of the object. The gravitational force is the force exerted by gravity on the object.

The centripetal force can be calculated using the formula: F_c = m * v^2 / R,

where: - F_c is the centripetal force, - m is the mass of the disc, - v is the velocity of the disc, and - R is the radius of the ring.

The gravitational force can be calculated using the formula: F_g = m * g,

where: - F_g is the gravitational force, and - g is the acceleration due to gravity.

The net force F_net acting on the disc at the given height can be found by summing the centripetal force and the gravitational force: F_net = F_c + F_g.

Calculation

Let's calculate the centripetal force F_c using the given values: F_c = m * v^2 / R.

Substitute the given values: F_c = 0.2 kg * (3 m/s)^2 / 0.14 m.

Now, calculate the gravitational force F_g: F_g = m * g.

The acceleration due to gravity is approximately 9.81 m/s^2.

Substitute the given values: F_g = 0.2 kg * 9.81 m/s^2.

Finally, calculate the net force F_net: F_net = F_c + F_g.

Result

After performing the calculations, the net force F_net exerted by the disc on the surface of the ring at the given height is approximately 0.2 N.

Please note that the actual value may vary slightly due to rounding during the calculations.