A 399 kg mass is brought close to a second

mass of 185 kg on a frictional surface with
coefficient of friction 1.5.
At what distance will the second mass begin to slide toward the first mass? The
acceleration of gravity is 9.8 m/s
2
and the
value of the universal gravitational constant
is 6.67259 × 10
−11
N · m2
/kg
2
.
Answer in units of mm

m1= 399kg, m2=185 kg

F(fr) = F =G•m1•m2/R²
μ•m2•g= G•m1•m2/R²
μ•g= G•m1/R²
R =sqrt(G•m1/ μ•g)

To find the distance at which the second mass will begin to slide toward the first mass, we need to determine the force of friction between the two masses.

The force of friction (Ff) can be calculated using the formula:

Ff = coefficient of friction * normal force

The normal force (Fn) is equal to the weight of the second mass, which can be calculated as:

Fn = mass * acceleration due to gravity

So, Fn = 185 kg * 9.8 m/s^2 = 1813 N

Substituting the given coefficient of friction (1.5) and the calculated normal force (1813 N) into the friction formula:

Ff = 1.5 * 1813 N = 2719.5 N

Now, let's analyze the forces acting on the second mass when it begins to slide. There are two forces: the force of gravity pulling the second mass downward (mg) and the force of friction opposing the motion. Since the second mass is not moving vertically, these two forces must balance each other:

Fg = Ff

mg = coefficient of friction * mass * acceleration due to gravity

Rearranging the equation to solve for distance (d):

d = Ff / (coefficient of friction * m * g)

Substituting the known values:

d = 2719.5 N / (1.5 * 185 kg * 9.8 m/s^2)

Calculating the distance:

d = 2719.5 N / (2727 N)

d ≈ 1 mm

Therefore, the second mass will begin to slide toward the first mass at a distance of approximately 1 mm.