A child (25 kg) walks along a 4 m plank of uniform density (m=8kg) that is laying with one end extending 1 m off the end of a dock.
1. Where is the ceanter of mass located in the plank?
2. What is the magnitude of the torque of the plank about the edge of the dock?
3. How far from the edge of the dock can the boy walk on the plank befopre the plank starts tipping into the water?
4. How could the boy test this conclusion without taking the risk of falling in the water?
To answer these questions, we need to understand the concept of center of mass and torque. Let's go step by step:
1. Finding the center of mass:
To find the center of mass of an object, we need to take into account its mass distribution. In this case, the plank is uniform, which means it has uniform mass distribution. The child is located 1 meter off the end of the dock, so we need to consider the masses separately.
We can calculate the center of mass using the formula:
Center of mass = (m1 × x1 + m2 × x2) / (m1 + m2)
Where,
m1 = mass of the plank = 8 kg
x1 = distance of the center of mass of the plank from the end on which the boy stands = 3 m (since the plank is 4 m long and the child is 1 m off the end)
m2 = mass of the child = 25 kg
x2 = distance of the child from the end of the dock = 1 m
Plugging these values into the formula:
Center of mass = (8 kg × 3 m + 25 kg × 1 m) / (8 kg + 25 kg)
Center of mass = (24 kg·m + 25 kg·m) / 33 kg
Center of mass ≈ 1.73 m
So, the center of mass of the plank and the child combined is approximately located 1.73 meters from the end of the dock.
2. Finding the torque:
Torque is a measure of how much a force applied on an object can cause it to rotate. In this case, we need to find the torque exerted on the plank about the edge of the dock.
The formula to calculate torque is:
Torque = force × distance
In this scenario, the force acting on the plank is due to the weight of the child and the plank itself. It can be calculated by multiplying the total mass by the acceleration due to gravity (9.8 m/s²).
Total mass = mass of the plank + mass of the child = 8 kg + 25 kg = 33 kg
Torque = total mass × acceleration due to gravity × distance
Distance = 4 m (length of the plank) - 1 m (distance of the child) = 3 m
Plugging in the values, we get:
Torque = 33 kg × 9.8 m/s² × 3 m
Torque = 970.2 N·m
Therefore, the magnitude of the torque of the plank about the edge of the dock is 970.2 N·m.
3. Determining the tipping point:
To find the distance from the edge of the dock where the plank starts tipping into the water, we need to consider the torque equilibrium. At the tipping point, the torque exerted by the weight of the child and the plank will be balanced by the torque exerted by the water acting as an upward force at one end.
In torque equilibrium, the torque on one side is equal to the torque on the other side.
Torque exerted by the child and the plank = Torque exerted by the water
(Total mass × acceleration due to gravity × distance) = (mass of the water × acceleration due to gravity × distance from the edge of the dock)
Here, total mass = mass of the plank + mass of the child = 8 kg + 25 kg = 33 kg
Simplifying the equation:
33 kg × 9.8 m/s² × (4 m - x) = m × 9.8 m/s² × x
Solving for x:
33 kg × 9.8 m/s² × 4 m - 33 kg × 9.8 m/s² × x = m × 9.8 m/s² × x
132 - 33x = 9.8x
42x = 132
x = 3.14 m
Therefore, the boy can walk approximately 3.14 meters from the edge of the dock before the plank starts tipping into the water.
4. Testing the conclusion without risk:
To test the conclusion without risking falling in the water, the boy can use a simple technique called "careful incremental movement." He can take small steps closer to the tipping point and observe if the plank starts to tip or not. By moving incrementally and checking the stability after each step, he can ensure his safety while testing the conclusion.