A cylindrical stone column of diameter 2R = 1.47 m and height H = 4.46 m is transported in standing position by a dolly. When the dolly accelerates or decelerates slowly enough, the column stands upright, but when the dolly’s acceleration magnitude exceed a critical value ac, the column topples over. (For a > +ac the column topples

backward; for a < −ac the column topples
forward.)

help please
Calculate the magnitude of the critical acceleration ac of the dolly. The acceleration of gravity is 9.8 m/s2.
Answer in units of m/s2.

To calculate the magnitude of the critical acceleration, ac, of the dolly, we need to determine the conditions under which the stone column will topple over.

Let's consider the forces acting on the stone column. One force is the gravitational force, given by F_gravity = m * g, where m is the mass of the stone column and g is the acceleration due to gravity (9.8 m/s^2). Another force is the torque caused by the acceleration of the dolly, which can cause the stone column to topple.

For a cylindrical column, the mass, m, can be calculated using the formula:
m = density * volume

Since the column is cylindrical, its volume, V, is given by:
V = π * (R^2) * H

Given that the diameter of the column is 2R = 1.47 m, we can find the radius, R, by dividing the diameter by 2.

R = 1.47 m / 2 = 0.735 m

Now we can calculate the volume:
V = π * (0.735^2) * 4.46 m^3

Next, we need to find the density of the stone column. Let's assume a typical value of 2700 kg/m^3 for the density of stone.

Now, we can calculate the mass of the column:
m = 2700 kg/m^3 * V

Once we have the mass, we can determine the critical acceleration ac by equating the torque of the dolly's acceleration to the torque caused by the gravitational force.

The torque caused by the dolly's acceleration is given by:
τ = m * (R/2) * ac

The torque caused by the gravitational force is given by:
τ_gravity = m * g * (R/2)

Setting τ = τ_gravity, we can solve for ac:
m * (R/2) * ac = m * g * (R/2)

Simplifying the equation, we find:
ac = g

Therefore, the magnitude of the critical acceleration, ac, is equal to the acceleration due to gravity, which is 9.8 m/s^2.