Block 1 is up an incline and is attacthed to block 2 by a rope, with m2 = 5.0 kg and è = 33°.If the coefficient of static friction between block #1 and the inclined plane is ìS = 0.24, what is the largest mass m1 for which the blocks will remain at rest?
To find the largest mass m1 for which the blocks will remain at rest, we need to consider the forces acting on the blocks and the conditions for equilibrium. Let's break it down step by step:
1. Draw a free body diagram for each block: Block 1 and Block 2.
2. Identify the forces acting on each block:
- Block 1: gravitational force (mg1), normal force (N1), static friction force (fs1), and tension in the rope (T).
- Block 2: gravitational force (mg2) and normal force (N2).
3. Determine the components of the forces:
- Block 1: The gravitational force can be split into two components, one parallel to the incline (mg1*sinθ) and one perpendicular to the incline (mg1*cosθ).
- Block 2: The gravitational force can be split into two components, one parallel to the incline (mg2*sinθ) and one perpendicular to the incline (mg2*cosθ).
4. Write down the equations for static equilibrium:
- ΣF = 0 in the x-direction (parallel to the incline) for Block 1: fs1 + T - mg1*sinθ = 0.
- ΣF = 0 in the y-direction (perpendicular to the incline) for Block 1: N1 - mg1*cosθ = 0.
- ΣF = 0 in the x-direction (parallel to the incline) for Block 2: T - mg2*sinθ = 0.
- ΣF = 0 in the y-direction (perpendicular to the incline) for Block 2: N2 - mg2*cosθ = 0.
5. Substitute the given values into the equations: m2 = 5.0 kg, θ = 33°, and μS = 0.24.
6. Solve the equations simultaneously to find the tension in the rope (T) and the normal forces (N1 and N2).
7. Use the value of the static friction force (fs1) to determine the maximum value of mg1 before sliding occurs:
- fs1 = μS * N1
8. Substitute the maximum value of fs1 into the equation ΣF = 0 in the x-direction for Block 1 to solve for mg1:
- fs1 + T - mg1*sinθ = 0
9. Solve for mg1, which will give you the largest mass for which the blocks will remain at rest.
By following these steps, you should be able to find the largest mass m1 for which the blocks will remain at rest.