Review Conceptual Example 7 before starting this problem. A uniform plank of length 5.0 m and weight 225 N rests horizontally on two supports, with 1.1 m of the plank hanging over the right support (see the drawing). To what distance x can a person who weighs 409 N walk on the overhanging part of the plank before it just begins to tip?
Where is the left support? We can't see your figure.
This is a static-equilibrium moment balance problem. When there is a zero moment about the right-most suport with zero force on the left-hand support, the plank will begin to tip. Remember to include the monent due to the weight of the plank.
To solve this problem, we need to consider the torque acting on the plank. Torque is the tendency of a force to rotate an object about an axis or pivot point. For the plank to remain in equilibrium, the torque acting on it must be zero.
Let's determine the forces acting on the plank first. There are three forces involved:
1. Weight of the plank (Wp): This force acts downward at the center of mass of the plank and has a magnitude of 225 N.
2. Weight of the person (Wp): This force acts downward at the point where the person is standing on the overhanging part of the plank. Its magnitude is given as 409 N.
3. Reaction force at the supports (Fr): This force acts upward at the points where the plank rests on the supports.
Next, we need to locate the pivot point or axis of rotation for the plank. Since the plank is in equilibrium, the net torque about any point must be zero. We can choose any point to calculate the torque, but it's convenient to choose the right support as the pivot point.
The torque due to each force can be calculated as the force multiplied by the perpendicular distance from the pivot point to the line of action of the force.
For the weight of the plank:
Torque due to weight of plank (τp) = Wp * (length of plank / 2)
= 225 N * (5.0 m / 2)
For the weight of the person:
Torque due to weight of person (τp) = Wp * (length of plank - x)
= 409 N * (5.0 m - x)
For the reaction force at the supports:
Torque due to reaction force (τr) = Fr * (length of plank - x - 1.1 m)
Now, we can set up the equation for torque equilibrium by summing up the torques acting on the plank:
τp + τp - τr = 0
Substituting the values of τp and τr:
225 N * (5.0 m / 2) + 409 N * (5.0 m - x) - Fr * (5.0 m - x - 1.1 m) = 0
Simplify and solve for x:
562.5 + 2045 - 409x - Fr * (3.9 - x) = 0
Now, we need to determine the reaction force Fr. Since the plank is in equilibrium, the sum of the vertical forces must be zero:
Fr + Wp + Wp = 0
Fr + 225 N + 409 N = 0
Solve for Fr.
Substitute the value of Fr into the equation for torque equilibrium and solve for x.
Once you have solved the equation, you will find the value of x, which represents the maximum distance a person can walk on the overhanging part of the plank without tipping it over.