Two uniform planks, each of mass m = 4.41 kg and length L = 2.11 m, are connected by a hinge at the top and by a chain of negligible mass attached at their centers, as shown in the figure. The assembly will stand upright, in the shape of an A, on a frictionless surface without collapsing. If the chain has the length 0.61 m, find each of the following:
a) the absolute value of the tension in the chain,
Tries 0/4
b) the absolute magnitude of the force on the hinge of each plank, and
Tries 0/4
c) the force of the ground on each plank.
To find the tension in the chain, we can use the concept of equilibrium. Since the assembly is standing upright on a frictionless surface without collapsing, the forces must be balanced at the hinge.
a) The tension in the chain:
We can start by analyzing the forces acting on one of the planks. The weight acts downwards, and the force at the hinge and the tension in the chain both act upwards. Since these forces are balanced, the magnitude of the tension in the chain will be equal to the magnitude of the force at the hinge.
To find the force at the hinge, we can consider the torques acting on the plank. Torque is the product of force and perpendicular distance from the axis of rotation. Since the axis of rotation is at the hinge, the perpendicular distance is simply half the length of the plank, L/2.
The torque due to the weight acting on the plank is zero because it acts at the center of the plank. The torque due to the tension in the chain can be calculated using the formula:
Torque = Force * Distance
The distance from the hinge to the center of the plank is also L/2. Hence, the torque due to the tension in the chain is Tension * (L/2). Since there is no angular acceleration, the torque due to the tension must balance the torque due to the weight:
(Tension * (L/2)) = (Weight * (L/2))
Note that the weight is given by the mass multiplied by the acceleration due to gravity (g).
Now we can solve for the tension in the chain. Substituting the given values for mass (m) and length (L), we can solve the equation:
(Tension * (2.11/2)) = (4.41 * 9.8 * (2.11/2))
Simplifying, we get:
Tension = (4.41 * 9.8)
Hence, the absolute value of the tension in the chain is approximately 43.14 N.
b) The force on the hinge of each plank:
Since the assembly is in equilibrium, the force on the hinge must be equal to the tension in the chain.
Therefore, the absolute magnitude of the force on the hinge of each plank is also approximately 43.14 N.
c) The force of the ground on each plank:
To find the force of the ground on each plank, we need to consider the vertical forces acting on the planks. The weight of each plank acts downwards, and the force at the hinge and the force of the ground both act upwards. Since these forces are balanced, the magnitude of the force of the ground on each plank will be equal to the magnitude of the weight acting on each plank.
The weight is given by the mass multiplied by the acceleration due to gravity (g).
Hence, the force of the ground on each plank is approximately (4.41 * 9.8) N.
To find the absolute value of the tension in the chain, we can consider the forces acting on the chain. Since the assembly is in equilibrium, the forces on each side of the chain must be equal in magnitude and opposite in direction.
Let's call the tension in the chain T. Since the chain is attached at the centers of the planks, the tension T will act vertically upward on each plank.
Now, let's consider one side of the chain. The chain will exert a force on the plank due to tension T. This force will act at the center of the plank and will be parallel to the plank.
Since the length of each plank is L = 2.11 m and the length of the chain is 0.61 m, the distance between the hinge and the center of the plank is (2.11 - 0.61)/2 = 0.75 m.
Since the forces are acting vertically upward, we can take the clockwise torques as positive and the counterclockwise torques as negative.
The torque due to the force exerted by the chain on the plank can be calculated as:
Torque = Force x Distance from hinge = T * 0.75
This torque must be balanced by the torque due to the weight of the plank. The weight of the plank can be calculated as:
Weight = mass x gravitational acceleration = 4.41 kg * 9.8 m/s^2
For equilibrium, the sum of the torques must be zero. So we have:
Torque due to chain force = Torque due to weight of plank
T * 0.75 = 4.41 kg * 9.8 m/s^2 * 2.11 m/2
Simplifying this equation, we can solve for T:
T = (4.41 kg * 9.8 m/s^2 * 2.11 m/2) / 0.75
Now you can calculate the value of T.