Final answer:
Kathy can walk towards the right until her weight creates a torque around trestle A that is equal to the sum of the torques from the plank's weight and the support at trestle B. We calculate this by setting up a torque balance equation and solving for the maximum distance Kathy can be from trestle A before the plank starts to tip.
Step-by-step explanation:
To determine how far Kathy can walk towards the right without toppling the plank, we need to use the principles of static equilibrium. For an object to be in static equilibrium, the sum of all forces and the sum of all torques (moments) acting on it must be zero.
Kathy's weight will act downwards at the point where she stands, and the plank's weight will act downwards at its center of gravity, which is in the middle of the plank. Assuming the plank is uniform, the center of gravity is 3m from either end. Since there is symmetry in the problem and trestles A and B are equidistant from the plank's center, both support an equal share of the plank's weight initially.
When Kathy stands 2.5 meters from trestle A, she is exerting a force equivalent to her weight, which needs to be balanced by the reactive forces at A and B to prevent the plank from rotating around either support. With her current position, there's a larger torque around trestle B because she is closer to it, so trestle A is providing less force than trestle B.
To find out how far Kathy can move to the right without the plank toppling, we can set up a torque balance equation about trestle A (taking trestle A as the pivot point). The plank will start to topple when the moment about trestle A due to Kathy's weight is equal to the sum of the moments due to the plank's weight and the support at trestle B.
Calculation
Kathy's maximum distance from trestle A can be calculated by the following torque balance at the tipping point:
Weight of Kathy × Distance from trestle A (to Kathy) = Weight of plank × Distance from trestle A (to center of plank) + Reactive force at trestle B × 3 m
Since she is already 2.5m away from A, the maximum additional distance she can move to the right before the plank begins to tip can be found by solving the equation for the distance from trestle A to Kathy. To prevent the plank from toppling the sum of torques around trestle A should not exceed the torque provided by the weight of the plank and the force at trestle B.