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
Well, this seems like a slippery situation! You're gonna need some math magic to solve this.
First, let's calculate the gravitational force acting between the masses. We can use the formula:
F = (G * m1 * m2)/r^2
Where F is the gravitational force, G is the universal gravitational constant, m1 and m2 are the masses, and r is the distance between them.
We want to find the distance at which the second mass starts sliding, so we need to consider the gravitational force and the frictional force acting on it.
The frictional force (Ff) can be calculated using:
Ff = μ * N
Where μ is the coefficient of friction and N is the normal force.
The normal force (N) can be calculated using:
N = m2 * g
Where g is the acceleration due to gravity.
To find the distance at which the second mass starts sliding, we need to equate the gravitational force with the frictional force:
(G * m1 * m2)/r^2 = μ * m2 * g
Now we can solve for the distance (r):
r^2 = (G * m1)/(μ * g)
r = √((G * m1)/(μ * g))
Plugging in the values:
r = √((6.67259 * 10^-11 N.m^2/kg^2 * 443 kg)/(0.5 * 9.8 m/s^2))
r ≈ 10513 mm
So, the second mass will begin sliding toward the first mass at a distance of approximately 10513 mm.
Remember, always be careful on slippery surfaces, even if they involve math!
To determine the distance at which the second mass will begin to slide toward the first mass, we need to consider the gravitational force between the two masses and the frictional force opposing the motion.
Let's break down the problem step by step:
Step 1: Calculate the gravitational force between the two masses.
The gravitational force can be calculated using the formula:
F = (G * m1 * m2) / r^2
where F is the gravitational force, G is the universal gravitational constant, m1 and m2 are the masses, and r is the distance between the centers of the masses.
Given:
m1 = 443 kg
m2 = 181 kg
G = 6.67259 × 10^-11 N · m^2 / kg^2
Let's plug in the values and calculate the gravitational force:
F = (6.67259 × 10^-11 N · m^2 / kg^2 * 443 kg * 181 kg) / r^2
Step 2: Determine the maximum static friction force.
The maximum static friction force can be calculated using the equation:
F_friction = μ * N
where F_friction is the frictional force, μ is the coefficient of friction, and N is the normal force.
Given:
μ = 0.5 (coefficient of friction)
The normal force can be calculated by balancing the forces vertically:
N = m2 * g
where g is the acceleration due to gravity.
Given:
g = 9.8 m/s^2
Step 3: Set the gravitational force equal to the maximum static friction force.
Since the object is on the verge of sliding, the gravitational force will be equal to the maximum static friction force.
F = F_friction
(6.67259 × 10^-11 N · m^2 / kg^2 * 443 kg * 181 kg) / r^2 = μ * m2 * g
Step 4: Solve for the distance (r).
Rearrange the equation to solve for r:
r^2 = (μ * m2 * g) / (6.67259 × 10^-11 N · m^2 / kg^2 * 443 kg * 181 kg)
Now let's calculate the value of r:
r = sqrt[(μ * m2 * g) / (6.67259 × 10^-11 N · m^2 / kg^2 * 443 kg * 181 kg)]
Step 5: Convert the distance to millimeters.
Since the given value of r is in meters, we need to convert it to millimeters.
1 meter = 1000 millimeters
Distance in millimeters = r * 1000
Now, let's calculate the distance in millimeters.
To determine at what distance the second mass will begin to slide toward the first mass, we need to consider the gravitational force and the frictional force acting on the second mass.
The gravitational force between two masses can be calculated using Newton's law of universal gravitation:
F_gravity = (G * m1 * m2) / r^2
Where:
- F_gravity is the gravitational force between the two masses,
- G is the universal gravitational constant (6.67259 × 10^-11 N * m^2 / kg^2),
- m1 and m2 are the masses of the two objects, and
- r is the distance between the two masses.
In this case, the first mass is 443 kg, and the second mass is 181 kg.
Now, let's consider the frictional force between the second mass and the surface. The frictional force can be calculated using:
F_friction = μ * F_normal
Where:
- F_friction is the frictional force,
- μ is the coefficient of friction (0.5 in this case),
- F_normal is the normal force (equal to the weight of the second mass), which can be calculated as F_normal = m2 * g, where g is the acceleration due to gravity (9.8 m/s^2).
To find the distance at which the second mass will begin to slide, we need to equate the gravitational force to the frictional force:
(G * m1 * m2) / r^2 = μ * F_normal
Substituting the known values, we have:
(6.67259 × 10^-11 N * m^2 / kg^2) * (443 kg) * (181 kg) / r^2 = (0.5) * (181 kg) * (9.8 m/s^2)
Now, we can solve this equation to find the distance (r). Rearranging the equation, we get:
r^2 = ((6.67259 × 10^-11 N * m^2 / kg^2) * (443 kg) * (181 kg)) / ((0.5) * (181 kg) * (9.8 m/s^2))
Taking the square root of both sides, we can find the value of r:
r = √(((6.67259 × 10^-11 N * m^2 / kg^2) * (443 kg) * (181 kg)) / ((0.5) * (181 kg) * (9.8 m/s^2)))
Calculating this using a calculator or by performing the above calculations step by step, we get:
r ≈ 643.58 m
The final step is to convert this distance to millimeters (mm):
1 meter = 1000 millimeters
Therefore,
r = 643.58 m * (1000 mm/1 m) ≈ 643580 mm
So, the distance at which the second mass will begin to slide toward the first mass is approximately 643580 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).