-4.44 A mass \(m_{1}=20.0 \mathrm{~kg}\) on a frictionless ramp is attached to a light string. The string passes over a frictionless pulley and is attached to a hanging mass \(m_{2}\). The ramp is at an angle of \(\theta=30.0^{\circ}\) above the horizontal. \(m_{1}\) moves up the ramp uniformly (at constant speed). Find the value of \(m_{2}\)

A free body diagram is a powerful tool in physics to visualize all the forces acting upon a single object. It's like drawing a map of all the force-vectors that comes into play for that object. In our textbook exercise, we draw separate diagrams for two masses: one on a ramp and one hanging.

In the diagram for the mass on the ramp, we see two forces: the weight of the mass acting directly downwards, represented by the force vector labeled as mg, and the tension in the string, which we'll call 'T'. This tension is what's keeping the mass from sliding down the ramp due to gravity. For the hanging mass, the forces are its weight (labeled as m2g), pulling down, and the tension 'T' pulling up. Creating a free body diagram is typically the first and a crucial step to solving problems in mechanics because it helps us visualize and thus better understand the forces at play.

A frictionless ramp and pulley system simplifies the complex real-world interactions into a more manageable problem by ignoring the effects of friction. This assumption allows us to focus solely on the gravitational and tension forces. When we say the ramp is frictionless, we're imagining a very smooth surface where the only force resisting the mass's movement up or down the ramp is its own weight component along the ramp, without any additional frictional forces.

Similarly, a frictionless pulley means that there is no resistance in the pulley's rotation, which in the real world could be caused by the axle or wheel rubbing against other surfaces. This ideal situation ensures that the tension in the rope is the same on either side of the pulley – a fact that simplifies calculations and lets students focus on the fundamental principles of mechanics without getting bogged down by complex friction calculations.

In physics problems, we often break down forces into components to simplify calculations. The component of gravitational force is a prime example. On an inclined plane, like our ramp, gravity pulls the mass both down towards the center of the Earth and along the plane.

Mathematically, we split the gravitational force into two parts: the component perpendicular to the ramp, which does not contribute to movement and is represented by \(m_{1}g\cos(\theta)\), and the parallel component, \(m_{1}g\sin(\theta)\), that causes the mass to slide down the ramp or requires tension in the string to resist this sliding. Understanding this concept is key to solving physics problems involving inclined planes, as it allows us to focus on the forces that actually cause movement.