Question

The figure below shows a beam of uniform density with a mass of 36.0 kg and a length ℓ = 4.40 m. It is suspended from a rope at a distance d = 1.20 m from its left end, while its right end is supported by a vertical column.

A beam of length ℓis suspended from a rope and supported by a vertical column. The rope is attached at a point that is a distance d from the left end of the beam. The column is beneath the right end of the beam.
(a) What is the tension (in N) in the rope?
Apply the equilibrium condition for torque about the right end of the beam. What is the torque due to the column at this point? Where does the gravitational force act? What is the torque due to this force? What is the torque due to the tension in the rope? What are the directions of the torques? N
(b) What is the magnitude of the force (in N) that the column exerts on the right end of the beam?

Answer 1

To find the tension in the rope, apply the equilibrium condition for torque about the right end of the beam. The tension in the rope is 230.88 N.

Step-by-step explanation:

To find the tension in the rope, we can apply the equilibrium condition for torque about the right end of the beam. First, let's determine the torques involved:

1. Torque due to the column at the right end: Since the column is supporting the right end of the beam, the torque due to the column is zero.

2. Torque due to the gravitational force: The gravitational force acts at the center of mass of the beam. In this case, the beam has uniform density, so the center of mass is at the midpoint of the beam. The torque due to the gravitational force can be calculated as T_g = (m*g*ℓ)/2, where m is the mass of the beam, g is the acceleration due to gravity, and ℓ is the length of the beam.

3. Torque due to the tension in the rope: We can assume that the tension in the rope acts perpendicular to the beam at the point of attachment. Therefore, the torque due to the tension in the rope is T_t = F_t*d, where F_t is the tension in the rope and d is the distance from the point of attachment to the right end of the beam.

Using these torques, we can set up the equilibrium condition for torque about the right end of the beam:

T_g + T_t = 0

Solving for the tension in the rope, we get:

F_t = -T_g/d

Substituting the values given in the problem, we have:

F_t = -(36.0 kg * 9.8 m/s^2 * 4.40 m) / 2.60 m = -230.88 N

Since tension is a positive quantity, the tension in the rope is 230.88 N.

Answer 2

To find the tension in the rope, we need to consider the equilibrium of forces and torques acting on the beam. The torque due to the column at the right end is zero, since the column exerts a normal force vertically and no torque is produced. The gravitational force acts at the center of mass of the beam, which creates a clockwise torque about the pivot point.

Step-by-step explanation:

To find the tension in the rope, we need to consider the equilibrium of forces and torques acting on the beam. Since the beam is in equilibrium, the sum of torques about any point must be zero. In this case, we can choose the right end of the beam as the pivot point.

The torque due to the column at the right end is zero, since the column exerts a normal force vertically and no torque is produced. The gravitational force acts at the center of mass of the beam, which is at a distance of 2.20 m from the right end. This force creates a clockwise torque about the pivot point.

The torque due to the tension in the rope is counterclockwise and is calculated by multiplying the tension by the perpendicular distance between the pivot point and the line of action of the tension, which is (4.40 m - 1.20 m) = 3.20 m

The sum of torques must be zero, so we can write:

Tension x 3.20 m = Gravitational force x 2.20 m

Now, we can solve for the tension in the rope:

Tension = (Gravitational force x 2.20 m) / 3.20 m