A 3.4 kg plant hangs from the bracket shown in the figure. The bracket has a mass of 0.65 kg , and its center of mass lies 9.0 cm from the wall. A single screw holds the bracket to the wall.
Find the horizontal tension force in the screw. Hint: Imagine that the bracket is slightly loose and pivoting about its bottom end. Assume the wall is frictionless.
To find the horizontal tension force in the screw, we can use the principle of torque. Torque is the rotational equivalent of force, and it is given by the equation:
τ = r * F * sin(θ)
where τ is the torque, r is the distance from the pivot point to the point where the force is applied, F is the force applied, and θ is the angle between the force and the line connecting the pivot point to the point where the force is applied.
In this case, the bracket is pivoting about its bottom end, so the torque is equal to zero. To calculate the torque, we need to consider all the torques acting on the system. There are two torques in this case: the torque due to the weight of the plant, and the torque due to the weight of the bracket itself.
Let's start by calculating the torque due to the weight of the plant. The weight of the plant can be calculated using the equation:
W = m * g
where W is the weight, m is the mass, and g is the acceleration due to gravity. Substituting the values, we get:
W = 3.4 kg * 9.8 m/s² = 33.32 N
Since the plant hangs vertically, the angle θ is 90 degrees. Hence, sin(θ) is 1. Now, we need to find the distance r from the pivot point to the point where the weight is acting. In this case, the distance is the distance between the center of mass of the plant and the pivot point of the bracket. Since this distance is not given directly, we need to calculate it.
The problem states that the center of mass of the bracket lies 9.0 cm from the wall. Since the plant hangs vertically, we can assume that the center of mass of the plant is in line with the center of mass of the bracket. Therefore, the distance r is equal to the distance between the center of mass of the bracket and the point where the screw is holding the bracket to the wall.
To find this distance, we need to subtract the distance between the pivot point and the wall from the distance between the center of mass of the bracket and the wall. Let's call the distance between the pivot point and the wall d.
We can solve for d using the equation:
d = distance between the center of mass of the bracket and the wall - 9.0 cm
Since the distance between the center of mass of the bracket and the wall is not given directly, we need to calculate it. We know that the center of mass of the bracket is located 9.0 cm from the wall. Therefore, the distance between the center of mass of the bracket and the wall is 0.0 cm.
Substituting this value into the equation, we get:
d = 0.0 cm - 9.0 cm = -9.0 cm
Now, we know that d is negative, which means that the pivot point is 9.0 cm to the left of the wall. Therefore, the distance r is equal to -9.0 cm.
Substituting the values into the torque equation, we get:
τ = (-9.0 cm) * (33.32 N) * (1) = -299.88 Ncm
Next, let's calculate the torque due to the weight of the bracket itself. The weight of the bracket can be calculated using the same equation:
W = m * g
Substituting the values, we get:
W = 0.65 kg * 9.8 m/s² = 6.37 N
In this case, the distance r is the distance between the center of mass of the bracket and the pivot point. We can assume that the center of mass of the bracket is located at its geometric center. Therefore, the distance r is equal to 9.0 cm.
Substituting the values into the torque equation, we get:
τ = (9.0 cm) * (6.37 N) * (1) = 57.33 Ncm
Since the torque is equal to zero, the sum of the torques is equal to zero:
(-299.88 Ncm) + (57.33 Ncm) = 0
To convert Ncm to Nm, we divide by 100:
(-299.88 Ncm) * (1 m / 100 cm) + (57.33 Ncm) * (1 m / 100 cm) = 0
(-2.9988 Nm) + (0.5733 Nm) = 0
-2.9988 Nm + 0.5733 Nm = 0
-2.4255 Nm = 0
To maintain the system in equilibrium, the horizontal tension force in the screw must be equal to the torque due to the weight of the plant and the torque due to the weight of the bracket itself. Therefore, the horizontal tension force in the screw is equal to:
T = -2.4255 Nm