(hrwc5p72) A mass m2 = 2.4 kg, on a 37.4° incline, is connected to a mass m1 = 6.9 kg, on a horizontal surface. The surfaces and the pulley are frictionless. If F = 18.9 N, what is the magnitude of the tension in the connecting cord?
18.9 N
To find the magnitude of the tension in the connecting cord, we can follow these step-by-step instructions:
Step 1: Draw a free-body diagram for each of the masses.
On mass m1 (6.9 kg) on the horizontal surface:
- There is only one force acting on m1, which is the tension force pulling to the right.
On mass m2 (2.4 kg) on the incline:
- There are two forces acting on m2: the gravitational force pulling downwards and the tension force pulling upwards and to the left.
Step 2: Apply Newton's second law (F = ma) in both the x and y directions for each mass.
For m1:
- In the x-direction: F = m1 * a1
- Since there is no acceleration in the x-direction (due to a frictionless surface), the tension force is equal to the force exerted by m1: T = F.
For m2:
- In the x-direction: T - m2 * g * sin(theta) = m2 * a2
- In the y-direction: m2 * g - T * cos(theta) = m2 * a3
- Since the pulley and surfaces are frictionless, the normal force and the gravitational force in the y-direction cancel each other out. Thus, m2 * g = T * cos(theta).
Step 3: Solve the system of equations to find the values of T, a2, and a3.
Using the equations from Step 2:
T - m2 * g * sin(theta) = m2 * a2 (Equation 1)
m2 * g - T * cos(theta) = m2 * a3 (Equation 2)
m2 * g = T * cos(theta) (Equation 3)
Substituting Equation 3 into Equation 2:
T * cos(theta) - T * cos(theta) = m2 * a3
0 = m2 * a3
a3 = 0
Substituting Equation 3 into Equation 1:
T - m2 * g * sin(theta) = m2 * a2
T - m2 * g * sin(theta) = m2 * 0
T = m2 * g * sin(theta)
Simplifying the equation:
T = 2.4 kg * 9.8 m/s^2 * sin(37.4°)
Step 4: Calculate the tension T.
Using a calculator or math software, compute the value of sin(37.4°) and substitute it into the equation:
T = 2.4 kg * 9.8 m/s^2 * 0.6018 (rounded to 4 decimal places)
T ≈ 14.11 N
Therefore, the magnitude of the tension in the connecting cord is approximately 14.11 N.
To solve this problem, we need to analyze the forces acting on the system and apply Newton's laws of motion. Here's how you can proceed:
1. Identify the forces: There are two main forces acting on the system: gravity and tension in the connecting cord.
- For m1 (the mass on the horizontal surface), the only force acting on it is its weight (mg), directed vertically downward.
- For m2 (the mass on the incline), there are two forces: the weight acting vertically downward (mg), and the tension in the connecting cord, which acts parallel to the incline.
2. Resolve the forces: Because the incline makes an angle of 37.4° with the horizontal, we need to resolve the weight (mg) and the tension force into their respective components.
- For m1, the weight component acting along the incline is m1 * g * sin(37.4°).
- For m2, the weight component acting along the incline is m2 * g * sin(37.4°), and the weight component acting perpendicular to the incline is m2 * g * cos(37.4°).
3. Apply Newton's second law: For m1, since it is on a horizontal surface and there is no net force in the horizontal direction, we have F - T = m1 * a (where T is the tension in the connecting cord and a is the acceleration of the system).
4. For m2, we can write the equation for the net force along the incline: T - m2 * g * sin(37.4°) = m2 * a.
5. Equate the accelerations: Since the masses are connected by the same cord, they have the same acceleration. Setting the two equations equal to each other, we get F - T = T - m2 * g * sin(37.4°).
6. Solve for T: Rearrange the equation to solve for T. T = (F + m2 * g * sin(37.4°)) / 2.
7. Plug in the given values: Substitute the known values (F = 18.9 N, m2 = 2.4 kg, and the angle = 37.4°) into the equation T = (F + m2 * g * sin(37.4°)) / 2 to find the magnitude of the tension in the connecting cord.
By following these steps, you should be able to calculate the magnitude of the tension in the connecting cord.