One end of a light uniform rod is attached to a wall by a...

One end of a light uniform rod is attached to a wall by a frictionless hinge. The rod is held in a horizontal position by a wire that runs from the other end of the rod to the wall. The wire has length $2.00 \mathrm{~m}$ and makes an angle of $30.0^{\circ}$ with the rod. A block with mass $m$ is suspended by a light rope attached to the middle of the rod. The transverse fundamental standing wave on the wire has frequency $f$. The mass $m$ is varied and for each value the frequency is measured. You plot $f^{2}$ versus $m$ and find that your data lie close to a straight line with slope $20.4 \mathrm{~kg}^{-1} \cdot \mathrm{s}^{-2}$ What is the mass of the wire? Assume that the change in the length of the wire when the tension changes is small enough to neglect.

Key Concepts

Tension in the String

The tension in a string or wire is a force that keeps it in equilibrium and determines the speed of wave propagation along it. When the tension is set by external forces, such as a gravitational force from an attached mass, it directly influences the frequency of the standing wave. Understanding this tension is key for linking the dynamic properties of waves to the static load applied.

Linear Mass Density

Linear mass density, defined as the mass per unit length of a string or wire, is an intrinsic property that affects how waves travel in the medium. In wave mechanics, knowing the linear mass density allows one to relate the tension in the string to the wave speed and the fundamental frequency, making it possible to extract material properties from experimental data such as the slope of a frequency-squared versus mass graph.

Standing Waves on a String

The study of standing waves involves understanding how waves reflect, interfere, and form nodes and antinodes along a medium under tension. The fundamental frequency of a string fixed at both ends is determined by its length, the tension in it, and its linear mass density. This concept underlies the relationship between the tension in the wire and the measurable wave frequency.

Static Equilibrium

Static equilibrium involves ensuring that the net forces and torques on a system are zero so that it remains at rest. In problems like this, the forces due to gravity, tension, and reaction forces must balance each other. Analyzing these forces using free?body diagrams and torque equations is crucial to relate different parts of the system and determine unknown quantities.

Torque and Lever Arm Analysis

Torque analysis is used to determine the conditions under which a body remains in rotational equilibrium. By choosing a pivot point and calculating the moments produced by the forces acting at various distances (lever arms), one can establish relationships between forces. This method is central to solving problems that involve rigid bodies connected by rods or hinges.

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