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 471 N walk on the overhanging part of the plank before it just begins to tip?
To determine the distance x that a person can walk on the overhanging part of the plank before it begins to tip, we need to find the point at which the sum of the torques acting on the plank is zero. This indicates that the plank is in equilibrium and is about to tip.
Let's break down the problem into steps:
Step 1: Identify the forces acting on the plank.
In this scenario, there are two forces acting on the plank:
1. The weight of the plank, which acts downward at its center. The weight of the plank can be calculated using the formula:
Weight of plank = mass of plank × acceleration due to gravity
= (225 N) / (9.8 m/s²) [assuming gravitational acceleration as 9.8 m/s²]
≈ 22.96 kg × 9.8 m/s² ≈ 225 N
2. The weight of the person, which acts downward at the point where the person is standing on the overhanging part of the plank. The weight of the person is given as 471 N.
Step 2: Calculate the total torque acting on the plank.
To calculate the torque, we need to define a point about which we will calculate it. Let's choose the left support as the point of reference. The torque acting on the plank due to its own weight can be calculated using the formula:
Torque of plank = Weight of plank × distance from center to left support
= (225 N) × (5.0 m / 2)
= 562.5 N·m
Similarly, the torque acting on the plank due to the person's weight can be calculated:
Torque of person = Weight of person × distance from left support to where the person stands
= (471 N) × (5.0 m - 1.1 m)
= 2150.5 N·m
Step 3: Set up the equilibrium equation.
The sum of the torques acting on the plank must be zero for the plank to be in equilibrium. Mathematically, this can be expressed as:
Torque of plank + Torque of person = 0
Substituting the known values, we have:
562.5 N·m + 2150.5 N·m = 0
2713 N·m = 0
Solving for the unknown distance x, we rearrange the equation and isolate x:
x = (Torque of person) / (Weight of plank)
= 2150.5 N·m / 225 N
≈ 9.56 m
Therefore, a person can walk on the overhanging part of the plank for approximately 9.56 meters before it just begins to tip.