To find the acceleration of each block and the tension in the cord, we can follow these steps:
Step 1: Draw a free-body diagram for each block.
For the block with mass m1:
- There is a normal force FN1 acting perpendicular to the inclined plane.
- The weight of the block mg1 is acting straight down.
- The force of kinetic friction fk1 is acting up the incline.
- The tension in the cord T is acting downward.
For the block with mass m2:
- There is a normal force FN2 acting upward.
- The weight of the block mg2 is acting straight down.
- The tension in the cord T is acting upward.
Step 2: Write down the equations for each block.
For the block with mass m1:
- In the direction parallel to the incline: m1 * a = fk1 - mg1 * sin(30°)
- In the direction perpendicular to the incline: FN1 = mg1 * cos(30°)
For the block with mass m2:
- In the direction perpendicular to the incline: FN2 = mg2
Step 3: Solve the system of equations.
First, solve for the tension T using the equation for the block with mass m1:
m1 * a = fk1 - mg1 * sin(30°)
Using the equation for kinetic friction: fk1 = μk1 * FN1
Substituting the value of FN1 and the given coefficient of friction (μk1 = 0.45), we get:
m1 * a = (0.45) * (mg1 * cos(30°)) - mg1 * sin(30°)
Next, substitute the value of FN2 from the equation for the block with mass m2 (FN2 = mg2) into the above equation to get:
m1 * a = (0.45) * (mg1 * cos(30°)) - mg1 * sin(30°) - mg2
Now, substitute the value of mg1 (mg1 = m1 * g) and mg2 (mg2 = m2 * g) into the equation:
m1 * a = (0.45) * (m1 * g * cos(30°)) - m1 * g * sin(30°) - m2 * g
Step 4: Solve for acceleration (a).
Rearrange the equation to solve for a:
a = [(0.45 * m1 * g * cos(30°)) - (m1 * g * sin(30°)) - (m2 * g)] / m1
Step 5: Solve for tension (T).
Substitute the value of a into the equation for the block with mass m1:
T = (0.45 * m1 * g * cos(30°)) - m1 * g * sin(30°) + m1 * a
Step 6: Calculate the values.
Substitute the given values into the equations and calculate the values:
- m1 = 8 kg
- m2 = 22 kg
- g = 9.8 m/s²
Calculate cos(30°) and sin(30°):
cos(30°) ≈ √3/2 ≈ 0.866
sin(30°) = 1/2 = 0.5
Now substitute the values into the equations and solve:
a = [(0.45 * 8 * 9.8 * 0.866) - (8 * 9.8 * 0.5) - (22 * 9.8)] / 8
T = (0.45 * 8 * 9.8 * 0.866) - (8 * 9.8 * 0.5) + (8 * a)
After calculations, you will find the acceleration of each block (a) and the tension in the cord (T).