Figure 12-54 a shows a horizontal uniform beam of mass mb...

Figure $12-54 a$ shows a horizontal uniform beam of mass $m_{b}$ and length $L$ that is supported on the left by a hinge attached to a wall and on the right by a cable at angle $\theta$ with the horizontal. A package of mass $m_{p}$ is positioned on the beam at a distance $x$ from the left end. The total mass is $m_{b}+m_{p}=61.22 \mathrm{kg}$ . Figure $12-54 b$ gives the tension $T$ in the cable as a function of the package's position given as a fraction $x / L$ of the beam length. The scale of the $T$ axis is set by $T_{a}=500 \mathrm{N}$ and $T_{b}=700 \mathrm{N}$ . Evaluate (a) angle $\theta,(\mathrm{b})$ mass $m_{b},$ and (c) mass $m_{p}$ .

Figure $12-54 a$ shows a horizontal uniform beam of mass $m_{b}$ and
length $L$ that is supported on the left by a hinge attached to a wall and on the right by a cable at angle $\theta$ with
the horizontal. A package of mass $m_{p}$ is positioned on the beam at a
distance $x$ from the left end. The total mass is $m_{b}+m_{p}=61.22 \mathrm{kg}$ . Figure $12-54 b$ gives the tension $T$ in the cable as a function of the
package's position given as a fraction $x / L$ of the beam length. The
scale of the $T$ axis is set by $T_{a}=500 \mathrm{N}$ and $T_{b}=700 \mathrm{N}$ . Evaluate (a)
angle $\theta,(\mathrm{b})$ mass $m_{b},$ and (c) mass $m_{p}$ .

Key Concepts

Static Equilibrium

Static equilibrium refers to the state in which a structure or body remains at rest because the vector sum of all forces and the sum of all moments (torques) about any point are zero. This concept is fundamental in analyzing mechanical systems to determine unknown forces, supports, and reactions.

Torque (Moment) Equilibrium

Torque, or moment of force, involves the rotational effect produced by a force acting at a distance from a pivot point. In equilibrium, the sum of torques about any chosen point must equal zero, which is instrumental in solving for unknown weights and forces acting on rigid bodies.

Free-Body Diagram

A free-body diagram is a simplified representation that isolates a body and shows all external forces and moments acting upon it. This graphical tool is essential in setting up the equations of equilibrium, enabling a clear visualization of the force interactions in systems like beams and cables.

Uniform Beam Properties

Uniform beam properties imply that the beam has a constant mass per unit length and that its weight can be assumed to act through its center of gravity. This simplification is crucial when determining the distribution of forces and moments along the beam.

Cable Tension and Force Resolution

In systems where a cable supports a load, the tension in the cable must be resolved into its horizontal and vertical components based on the angle of the cable relative to the horizontal. Understanding how to resolve this tension is key to applying static equilibrium conditions and determining unknown forces in the system.

Transcript

Please give Ace some feedback