The weight of a car with a mass of m = 1.2 tons, moving at a speed of v = 79.2 km / h on a concave bridge

The weight of a car with a mass of m = 1.2 tons, moving at a speed of v = 79.2 km / h on a concave bridge, is P = 24960 N at the lowest point of the bridge. What is the radius of curvature of the bridge? Indicate in the explanatory drawing the forces acting on the vehicle at the lowest point of the bridge.

P = 24960 N.

m = 1.2 t = 1200 kg.

g = 10 m / s2.

V = 79.2 km / h = 22 m / s.

R -?

The weight of the car P is the force with which it presses on the bridge.

Two forces act on the car: gravity Ft directed vertically downward, axle reaction force N directed vertically upward.

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

Since the bridge is concave, then for projections on the vertical axis 2 Newton’s law will take the form: m * a = – Fт + N.

N = m * a + Fт.

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

N = m * a + m * g.

The centripetal acceleration a is expressed by the formula: a = V2 / R.

N = m * V2 / R + m * g.

m * V2 / R = N – m * g.

R = m * V2 / (N – m * g).

According to Newton’s 3 laws, the force N with which the bridge presses on the car is equal to the force with which the car presses on the bridge Р: N = P.

R = m * V2 / (P – m * g).

R = 1200 kg * (22 m / s) 2 / (24960 N – 1200 kg * 10 m / s2) = 44.8 m.

Answer: the radius of curvature of the bridge should be R = 44.8 m.