A beam resting on two pivots has a length of L = 9.00 m and mass M =89.0 kg. The pivot under the left end exerts a normal force n1 on the beam, and the second pivot placed a distance 4.00 m from the left end exerts a normal force n2. A woman of mass m= 48.0 kg steps onto the left end of the beam and begins walking to the right as in the figure. The goal is to find the woman’s position when the beam begins to tip. (d) Use the torque equilibrium equation, with torques computed around the second pivot point and find the woman’s position when the beam is about to tip
To find the woman's position when the beam begins to tip, we need to use torque equilibrium equation and compute torques around the second pivot point.
Torque is the product of the force and the perpendicular distance from the pivot point to the line of action of the force. In this case, the torque equilibrium equation can be written as:
Στ = 0,
where Στ is the sum of all torques acting on the beam.
Let's analyze the forces and their torques:
1. The weight of the beam (W_beam) acts downward at its center of mass. The torque contributed by this force is zero since the distance from the center of mass to the second pivot point is zero.
2. The normal force n1 exerted by the first pivot at the left end of the beam can be split into a vertical component (n1⊥) and a horizontal component (n1∥). The vertical component doesn't contribute to the torque since its distance from the second pivot is zero. The horizontal component, however, does contribute to the torque.
3. The weight of the woman (W_woman) acts downward at her center of mass. As the woman walks to the right, her weight creates a torque that tends to tip the beam.
4. The normal force n2 exerted by the second pivot creates an upward force that counteracts the torque produced by the woman's weight. The distance from n2 to the second pivot is given as 4.00 m.
Using these forces and their torques, we can write the torque equilibrium equation as:
(n1∥ × 0) + (W_woman × d) - (n2 × 4.00) = 0,
where n1∥ is the horizontal component of the normal force n1, d is the displacement of the woman from the second pivot, and n2 is the normal force exerted by the second pivot.
Solving the equation for d will give us the woman's position when the beam begins to tip.