Question

forces review assignment

1. draw the free body diagrams for each mass in an atwoods machine. (see picture.)

Explanation:

Identify the system and masses

We analyze the Atwood's machine consisting of two masses, \(M\) and \(m\), connected by a light, inextensible string passing over a frictionless pulley. From the diagram, mass \(M\) is visually larger than mass \(m\), implying \(M > m\).

Analyze forces on mass M

Using the Free Body Diagram concept, we identify the forces acting on the larger mass \(M\):

1. Gravity pulling downward: \(F_{g,M} = Mg\)
2. Tension from the string pulling upward: \(T\)

Since \(M > m\), mass \(M\) accelerates downward, meaning the downward gravitational force is larger than the upward tension force: \(Mg > T\).

Analyze forces on mass m

Using the Free Body Diagram concept, we identify the forces acting on the smaller mass \(m\):

1. Gravity pulling downward: \(F_{g,m} = mg\)
2. Tension from the string pulling upward: \(T\)

Since mass \(m\) accelerates upward, the upward tension force is larger than the downward gravitational force: \(T > mg\).

Compare force vector lengths

For a correct visual representation on the grid:

• The tension \(T\) is equal in magnitude for both masses because they are connected by the same light string.
• For mass \(M\), the downward vector \(F_{g,M}\) must be longer than the upward vector \(T\).
• For mass \(m\), the upward vector \(T\) must be longer than the downward vector \(F_{g,m}\).
• Thus, the relative lengths of the force vectors are: \(F_{g,M} > T > F_{g,m}\).

Answer:

To draw the free body diagrams for each mass in the Atwood's machine:

1. For Mass \(M\) (larger mass):

• Draw a downward arrow from the center dot labeled \(F_{g,M}\) (or \(Mg\)) representing gravity.
• Draw an upward arrow from the center dot labeled \(T\) representing tension.
• Make the downward arrow \(F_{g,M}\) longer than the upward arrow \(T\) (e.g., \(F_{g,M}\) is 3 grid units long, and \(T\) is 2 grid units long).

1. For Mass \(m\) (smaller mass):

• Draw a downward arrow from the center dot labeled \(F_{g,m}\) (or \(mg\)) representing gravity.
• Draw an upward arrow from the center dot labeled \(T\) representing tension.
• Make the upward arrow \(T\) longer than the downward arrow \(F_{g,m}\) (e.g., \(T\) is 2 grid units long, and \(F_{g,m}\) is 1 grid unit long).