A beam of mass M and length L is supported horizontally at...

A beam of mass $M$ and length $L$ is supported horizontally at its ends by two cables making angles $\theta$ and $\phi$ with the horizontal ceiling (Fig. 11.41$)$ . (a) Show that if the beam is uniform, these two angles must be equal and the tensions in the cables must also be equal. (b) Suppose now that the center of gravity is 3$L / 4$ from the left end of the beam. Show that the angles are not completely independent but must obey the equation $\tan \theta=3 \tan \phi .$

A beam of mass $M$ and length $L$ is supported horizontally at its ends by two cables making angles $\theta$ and $\phi$ with the horizontal ceiling (Fig. 11.41$)$ . (a) Show that if the beam is uniform, these two angles must be equal and the tensions in the cables must also be equal. (b) Suppose now that the center of gravity is 3$L / 4$ from the left end of the beam. Show that the angles are not completely
independent but must obey the equation $\tan \theta=3 \tan \phi .$

Key Concepts

Static Equilibrium

Static equilibrium refers to the condition where the net force and net torque acting on a system are both zero, ensuring the object remains at rest. In the context of beams supported by cables, it means that the sum of the vertical and horizontal force components, as well as the sum of torques about any pivot point, must cancel out.

Force Resolution

Force resolution involves breaking down forces into their horizontal and vertical components using trigonometric functions, especially when forces are applied at angles. This is crucial for analyzing systems such as a beam held by angled cables, as it allows the establishment of independent equilibrium equations for each direction.

Torque and Moment Equilibrium

Torque, or moment of force, is the product of a force and its lever arm, and it causes rotation around a pivot point. Moment equilibrium requires that the sum of all torques about any chosen pivot must be zero. This concept is essential in problems involving beams to balance the rotational effects of forces, particularly when the center of mass is not at the geometric center.

Center of Mass

The center of mass is the effective point at which the mass of an object is considered to be concentrated. In static equilibrium problems, such as a beam with a shifted center of gravity, the location of the center of mass affects the distribution of weight and the resulting torques, influencing the force and tension conditions required for balance.

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