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 408 N walk on the overhanging part of the plank before it just begins to tip?
2.27
To determine the distance x a person can walk on the overhanging part of the plank before it begins to tip, we need to calculate the torque exerted by the person's weight and compare it to the torque exerted by the weight of the plank.
The torque exerted by a force is given by the formula: torque = force × distance
In this case, the weight of the person creates a torque that tries to tip the plank. The torque exerted by the person's weight (T_person) is given by: T_person = weight_person × distance_person
The weight of the plank creates a torque in the opposite direction. The torque exerted by the weight of the plank (T_plank) is given by: T_plank = weight_plank × distance_plank
We know the weight of the person (408 N), the weight of the plank (225 N), the length of the plank (5.0 m), and the distance the plank overhangs the right support (1.1 m).
First, let's calculate the torque exerted by the person's weight:
T_person = weight_person × distance_person
T_person = 408 N × 1.1 m
T_person = 448.8 N·m
Next, let's calculate the torque exerted by the weight of the plank:
T_plank = weight_plank × distance_plank
T_plank = 225 N × (5.0 m - 1.1 m)
T_plank = 225 N × 3.9 m
T_plank = 877.5 N·m
To prevent the plank from tipping, the torque created by the person's weight should not exceed the torque created by the weight of the plank.
Therefore, the maximum distance (x) the person can walk on the overhanging part of the plank before it begins to tip is the distance from the right support to where the torque exerted by the person's weight equals the torque exerted by the weight of the plank.
Setting the two torques equal to each other:
T_person = T_plank
408 N × x = 225 N × (5.0 m - 1.1 m)
408 N × x = 877.5 N·m
x = 877.5 N·m / 408 N
x ≈ 2.15 m
Therefore, the person can walk approximately 2.15 meters on the overhanging part of the plank before it just begins to tip.
To determine the maximum distance x that a person can walk on the overhanging part of the plank before it starts to tip, we need to consider the equilibrium condition.
In this scenario, the plank will start to tip when the torque exerted by the person's weight on one side of the fulcrum exceeds the torque exerted by the weight of the overhanging part of the plank on the other side of the fulcrum.
First, we need to calculate the torque exerted by the weight of the person on the overhanging part of the plank. The formula for torque is τ = F × d, where τ is the torque, F is the force, and d is the perpendicular distance from the point of rotation (fulcrum) to the line of action of the force.
In this case, the perpendicular distance is the distance from the fulcrum to the end of the plank, which is 1.1 m.
The torque exerted by the person is τ_person = F_person × d_person = (408 N) × (1.1 m) = 448.8 N·m.
To maintain equilibrium, the torque exerted by the weight of the overhanging portion of the plank must equal the torque exerted by the person. Let's call the weight of the overhanging part of the plank W_plank.
The torque exerted by the weight of the overhanging part of the plank is τ_plank = W_plank × d_plank, where d_plank is the perpendicular distance from the fulcrum to the center of mass of the overhang.
Since the plank is uniform, the center of mass of the overhang is located at its midpoint. Therefore, the perpendicular distance from the fulcrum to the center of mass of the overhang is half of the length of the overhang, which is (1.1 m) / 2 = 0.55 m.
So, the torque exerted by the weight of the overhanging part of the plank is τ_plank = (W_plank) × (0.55 m).
According to the equilibrium condition, the torque exerted by the person (τ_person) on one side of the fulcrum should be equal to the torque exerted by the weight of the overhanging part of the plank (τ_plank) on the other side of the fulcrum.
Therefore, we have the equation:
τ_person = τ_plank
(448.8 N·m) = (W_plank) × (0.55 m)
Now, we can solve for W_plank:
W_plank = (448.8 N·m) / (0.55 m)
W_plank ≈ 816 N
The weight of the overhanging part of the plank is approximately 816 N.
Now that we know the weight of the overhanging part of the plank, we can calculate the maximum distance x by using the equation τ = F × d and considering the torque exerted by the person (τ_person) on the overhanging part of the plank.
τ_person = F_person × d_person
(408 N) × (x) = (816 N) × (1.1 m)
Now, let's solve for x:
408 N × x = 816 N × 1.1 m
408 N × x = 897.6 N·m
x ≈ 2.2 m
Therefore, a person who weighs 408 N can walk approximately 2.2 m on the overhanging part of the plank before it just begins to tip.