Objects of masses m1 = 4.00 kg and m2 = 9.00 kg are connected by a light string that passes over a frictionless pulley as in the figure below. The object m1 is held at rest on the floor, and m2 rests on a fixed incline of θ = 39.0°. The objects are released from rest, and m2 slides 1.80 m down the slope of the incline in 4.50 s.
acceleration is .1778
Determine the tension in the string. (Enter the magnitude only.)
(c) Determine the coefficient of kinetic friction between m2 and the incline
To determine the tension in the string, we can use the concept of Newton's second law:
For object m1:
Sum of forces = m1 * acceleration
For object m2:
Sum of forces = m2 * acceleration
Since the two objects are connected by a string, the tension in the string will be the same for both objects.
1. Find the acceleration:
Given that m2 slides 1.80 m down the slope in 4.50 s, we can use the formula of motion:
s = ut + (1/2)at^2
where s is the distance traveled, u is the initial velocity, t is the time taken, and a is the acceleration.
Here, the initial velocity (u) is 0 because the objects are released from rest. Therefore, the formula reduces to:
s = (1/2)at^2
Substituting the values, we have:
1.80 = (1/2) * a * (4.50)^2
Solving for a, we get:
a = 0.1778 m/s^2
2. Calculate the tension in the string:
We know that the tension in the string is the same for both objects, so we can choose either m1 or m2 to calculate the tension. Let's use m1.
Since m1 is at rest on the floor, the tension in the string is equal to the weight of m1.
Tension = m1 * g
= 4.00 kg * 9.8 m/s^2 (acceleration due to gravity)
Tension = 39.2 N (approximately)
Therefore, the tension in the string is approximately 39.2 N.
(c) To determine the coefficient of kinetic friction between m2 and the incline, we need to consider the forces acting on m2.
The forces acting on m2 are:
- The weight of m2 (m2 * g) acting vertically downward
- The tension in the string (T) acting parallel to the incline but upwards
- The force of kinetic friction (fk) opposing the motion of m2 sliding down the incline
Since m2 is sliding down the slope, the force of kinetic friction can be calculated as:
fk = μk * N
where μk is the coefficient of kinetic friction and N is the normal force.
The normal force, N, can be determined by considering the component of m2's weight perpendicular to the incline:
N = m2 * g * cos(θ)
where θ is the angle of the incline (39.0°).
Now we can substitute the values into the formula to solve for the coefficient of kinetic friction:
fk = μk * N
The weight of m2 can be calculated as:
m2 * g = 9.00 kg * 9.8 m/s^2
The normal force N can be calculated as:
N = 9.00 kg * 9.8 m/s^2 * cos(39.0°)
Finally, substitute these values into the formula and solve for the coefficient of kinetic friction (μk).
To determine the tension in the string, we can use Newton's second law and the concept of acceleration. Here's how to calculate it step-by-step:
1. Start by finding the acceleration of the system. From the given information, we know that the object m2 slides down the incline, so we can use the kinematic equation:
Δx = v₀t + (1/2)at²
where Δx is the displacement (1.80 m), v₀ is the initial velocity (0 m/s), t is the time (4.50 s), and a is the acceleration (unknown).
Substitute the known values into the equation and solve for a:
1.80 m = 0 m/s * 4.50 s + (1/2) * a * (4.50 s)²
Simplify the equation:
1.80 m = (1/2) * a * (4.50 s)²
Divide through by (1/2) * (4.50 s)²:
a = (1.80 m) / [(1/2) * (4.50 s)²]
Calculate the value of a:
a ≈ 0.1778 m/s²
2. Since the masses are connected by a light string passing over a frictionless pulley, the tension in the string is the same for both objects. We can find the tension by applying Newton's second law to each object separately.
For m1:
Sum of forces on m1 = m1 * a₁
The net force on m1 is the tension T, so we have:
T - m1 * g = m1 * a₁
For m2:
Sum of forces on m2 = m2 * a₂
The only force acting on m2 is its weight component down the incline, so we have:
m2 * g * sin(θ) - T = m2 * a₂
3. Notice that the acceleration a₁ of m1 is 0 since it is held at rest. Combined with the fact that g is the same for both objects, the equations simplify to:
T = m2 * g * sin(θ)
4. Substitute the known values into the equation and solve for T:
m2 = 9.00 kg
g ≈ 9.8 m/s²
θ = 39.0°
T = (9.00 kg) * (9.8 m/s²) * sin(39.0°)
5. Calculate the value of T:
T ≈ 52.057 N
Therefore, the tension in the string is approximately 52.057 N.
Next, we can determine the coefficient of kinetic friction between m2 and the incline. Here's how to calculate it step-by-step:
1. The force of kinetic friction (fk) is equal to the normal force (N) multiplied by the coefficient of kinetic friction (μk):
fk = μk * N
2. The normal force N can be determined using the equation:
N = m2 * g * cos(θ)
3. Substitute the known values into the equation and solve for N:
m2 = 9.00 kg
g ≈ 9.8 m/s²
θ = 39.0°
N = (9.00 kg) * (9.8 m/s²) * cos(39.0°)
4. Calculate the value of N:
N ≈ 68.829 N
5. The force of kinetic friction can be determined using the equation:
fk = m2 * g * μk * cos(θ)
6. Notice that the force of kinetic friction is in the opposite direction of motion. Hence, we have:
fk = m2 * a₂
7. Since the acceleration a₂ is given as 0.1778 m/s², we can solve for μk:
m2 * a₂ = m2 * g * μk * cos(θ)
μk = a₂ / (g * cos(θ))
8. Substitute the known values into the equation and calculate μk:
a₂ = 0.1778 m/s²
g ≈ 9.8 m/s²
θ = 39.0°
μk = (0.1778 m/s²) / [(9.8 m/s²) * cos(39.0°)]
9. Calculate the value of μk:
μk ≈ 0.0457
Therefore, the coefficient of kinetic friction between m2 and the incline is approximately 0.0457.