A mass m1 = 18.7 kg on a frictionless ramp is attached to a light string. The string passes over a frictionless pulley and is attached to a hanging mass m2. The ramp is at an angle of θ = 23.7° above the horizontal. m1 moves up the ramp uniformly (at constant speed). Find the value of m2.
To find the value of m2, we need to analyze the forces acting on the system.
1. Draw a free-body diagram for mass m1 on the ramp. The forces acting on m1 are:
- The weight force (mg1) acting straight downward.
- The normal force (N) acting perpendicular to the ramp.
- The tension force (T) in the string, which is directed up the ramp.
2. Decompose the weight force (mg1) into components parallel and perpendicular to the ramp. The component parallel to the ramp is mg1*sin(θ), and the component perpendicular to the ramp is mg1*cos(θ).
3. Since m1 is moving up the ramp uniformly at constant speed, the forces parallel to the ramp must be balanced. Therefore, the tension force T is equal to the component of the weight force parallel to the ramp. This gives us T = mg1*sin(θ).
4. Now, let's consider mass m2 hanging vertically. The forces acting on m2 are:
- The weight force (mg2) acting straight downward.
- The tension force (T) in the string, which is directed downward.
5. Similar to step 3, the tension force T in the string is equal to the weight force mg2. This gives us T = mg2.
6. Since we know that T = mg1*sin(θ) and T = mg2, we can set these two expressions equal to each other:
mg1*sin(θ) = mg2.
7. Now we can substitute the given values of m1 = 18.7 kg and θ = 23.7° into the equation:
18.7 kg * sin(23.7°) = mg2.
8. Calculate the right side of the equation:
(18.7 kg) * (sin(23.7°)) ≈ 7.88 kg.
Therefore, the value of m2 is approximately 7.88 kg.