A 443 kg mass is brought close to a second
mass of 181 kg on a frictional surface with
coefficient of friction 0.5.
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= 443 kg, m2=181 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 compare the gravitational force between the two masses with the maximum frictional force that can act on the second mass.
Let's start by calculating the gravitational force between the two masses using Newton's law of universal gravitation:
F_grav = G * (m1 * m2) / r^2
Where:
- F_grav is the gravitational force between the masses,
- G is the universal gravitational constant,
- m1 and m2 are the masses of the objects, and
- r is the distance between the centers of the masses.
In this case, m1 is the mass that is brought close to the second mass (443 kg), and m2 is the second mass (181 kg). The value of the universal gravitational constant is given as 6.67259 × 10^-11 N·m^2/kg^2.
Next, we need to calculate the maximum frictional force that can act on the second mass. The maximum frictional force can be determined by multiplying the coefficient of friction (0.5) by the normal force between the objects. The normal force is equal to the weight of the second mass, which is given by:
F_normal = m2 * g
Where g is the acceleration due to gravity (9.8 m/s^2).
Finally, we equate the gravitational force and the maximum frictional force to find the distance at which the second mass starts to slide:
G * (m1 * m2) / r^2 = μ * m2 * g
Where μ is the coefficient of friction.
To solve for r, we rearrange the equation:
r = √(G * m1 / (μ * g))
Now we can substitute the given values into the equation and calculate the result. Remember to convert the final answer to units of mm.
r = √((6.67259 × 10^-11 N·m^2/kg^2 * 443 kg) / (0.5 * 181 kg * 9.8 m/s^2))
After performing the calculations, the distance r is obtained.