011 10.0 points
Masses 18 kg and 9 kg are connected by a light string that passes over a frictionless pulley as shown in the figure.
The acceleration of gravity is 9.8 m/s2 . 18 kg
9 kg
If the 18 kg mass, initially held at rest on the table, is released and moves 1.7 m in 1.4 s, determine the coefficient of kinetic friction between it and the table.
To determine the coefficient of kinetic friction between the 18 kg mass and the table, we can use the equation for calculating acceleration:
acceleration = (net force) / (mass)
We can also express the net force as the difference between the tension in the string and the force of friction:
net force = tension - force of friction
Since the masses are connected by a light string that passes over a frictionless pulley, the tension in the string pulling the 9 kg mass is the same as the tension in the string pulling the 18 kg mass. Therefore, we can write the equation for the net force as:
net force = (tension) - (force of friction)
The force of friction can be expressed as the product of the coefficient of kinetic friction and the normal force. The normal force is equal to the weight of the 18 kg mass:
force of friction = (coefficient of kinetic friction) * (normal force)
The weight, in this case, is the mass multiplied by the acceleration due to gravity:
weight = mass * acceleration due to gravity
Now we can plug in the given values to find the coefficient of kinetic friction:
mass = 18 kg
acceleration due to gravity = 9.8 m/s^2
distance moved = 1.7 m
time taken = 1.4 s
First, we can calculate the acceleration of the 18 kg mass using the formula:
acceleration = (distance moved) / (time taken)
acceleration = 1.7 m / 1.4 s = 1.2143 m/s^2 (rounded to four decimal places)
Next, we calculate the weight of the 18 kg mass:
weight = mass * acceleration due to gravity
weight = 18 kg * 9.8 m/s^2 = 176.4 N
Substituting this value into the equation for the force of friction:
force of friction = (coefficient of kinetic friction) * (normal force)
force of friction = (coefficient of kinetic friction) * (weight)
Since the mass is moving, the force of friction is opposing its motion. Therefore, the force of friction is equal to the mass multiplied by the acceleration:
force of friction = mass * acceleration
force of friction = 18 kg * 1.2143 m/s^2 = 21.8574 N (rounded to four decimal places)
Now we can equate the net force and solve for the tension:
net force = (tension) - (force of friction)
Since the net force is equal to the mass multiplied by the acceleration:
(mass) * (acceleration) = (tension) - (force of friction)
(18 kg) * (1.2143 m/s^2) = (tension) - 21.8574 N
Tension = (18 kg) * (1.2143 m/s^2) + 21.8574 N
Tension = 43.8566 N (rounded to four decimal places)
Since the tension is the same for both masses, the 9 kg mass also experiences a tension of 43.8566 N.
Now we can calculate the coefficient of kinetic friction using the force of friction and the weight:
force of friction = (coefficient of kinetic friction) * (weight)
21.8574 N = (coefficient of kinetic friction) * 176.4 N
Coefficient of kinetic friction = 21.8574 N / 176.4 N ≈ 0.124 (rounded to three decimal places)
Therefore, the coefficient of kinetic friction between the 18 kg mass and the table is approximately 0.124.
To determine the coefficient of kinetic friction between the 18 kg mass and the table, we need to analyze the forces acting on the mass.
First, let's calculate the net force acting on the 18 kg mass. We'll use Newton's second law of motion, which states that the net force (F_net) on an object is equal to its mass (m) multiplied by its acceleration (a):
F_net = m * a
In this case, the only force acting on the 18 kg mass is the tension in the string. The tension (T) in the string can be calculated using the following equation:
T = m_1 * g - m_2 * g
Where m_1 and m_2 are the masses hanging on both sides of the pulley, and g is the acceleration due to gravity. In this case, m_1 = 18 kg, m_2 = 9 kg, and g = 9.8 m/s^2.
T = (18 kg * 9.8 m/s^2) - (9 kg * 9.8 m/s^2) = 176.4 N - 88.2 N = 88.2 N
Now, we can calculate the acceleration (a) of the 18 kg mass using the equation:
a = (2*distance) / (time^2)
In this case, the distance is given as 1.7 m and the time is given as 1.4 s.
a = (2 * 1.7 m) / (1.4 s)^2 = 2.43 m/s^2
Finally, we can substitute the calculated values for F_net and m into the formula for kinetic friction:
F_friction = μ * F_normal
Where F_friction is the force of friction, μ is the coefficient of kinetic friction, and F_normal is the normal force exerted by the table on the 18 kg mass.
The normal force (F_normal) is equal to the weight of the 18 kg mass, which can be found using the equation:
F_normal = m * g
F_normal = 18 kg * 9.8 m/s^2 = 176.4 N
Substituting the values into the formula, we get:
F_friction = μ * 176.4 N
Since the 18 kg mass is moving, the force of friction is equal to the net force acting on it. Therefore:
F_friction = F_net = m * a = 18 kg * 2.43 m/s^2 = 43.74 N
Now we can solve for the coefficient of kinetic friction (μ):
43.74 N = μ * 176.4 N
μ = 43.74 N / 176.4 N = 0.248
Therefore, the coefficient of kinetic friction (μ) between the 18 kg mass and the table is approximately 0.248.