A 1300-N uniform boom at phi = 64.0° to the horizontal is supported by a cable at an angle θ = 26.0° to the horizontal as shown in the figure below. The boom is pivoted at the bottom, and an object of weight w = 2250 N hangs from its top.
To find the tension in the cable and the horizontal and vertical components of the force exerted by the boom on the pivot, we can analyze the forces acting on the boom.
Let's break down the problem step by step:
Step 1: Draw a free-body diagram of the boom.
- Draw a horizontal line to represent the ground.
- Draw a vertical line to represent the cable attached to the boom.
- Draw a line perpendicular to the boom to represent the force exerted by the pivot on the boom.
- Label the angles given in the problem statement.
- Label the known forces: the weight of the boom (1300 N), the weight of the object (2250 N), and the tension in the cable.
Step 2: Resolve the weight of the boom into horizontal and vertical components.
- The weight of the boom acts vertically downward.
- Resolve the weight into its vertical component (w_y) and horizontal component (w_x).
- Use trigonometry to calculate w_y = weight of the boom * sin(phi).
- Use trigonometry to calculate w_x = weight of the boom * cos(phi).
Step 3: Resolve the tension in the cable into horizontal and vertical components.
- The tension in the cable acts along an angle θ to the horizontal.
- Resolve the tension into its vertical component (T_y) and horizontal component (T_x).
- Use trigonometry to calculate T_y = tension in the cable * sin(theta).
- Use trigonometry to calculate T_x = tension in the cable * cos(theta).
Step 4: Apply the principle of equilibrium.
- The boom is in equilibrium, which means the sum of the forces in both the horizontal and vertical directions is zero.
- In the horizontal direction, the forces are T_x (from the cable) and -w_x (from the weight of the boom). Set the sum to zero: T_x - w_x = 0.
- In the vertical direction, the forces are T_y (from the cable), -w_y (from the weight of the boom), and the weight of the object. Set the sum to zero: T_y - w_y - weight of the object = 0.
Step 5: Solve the equations.
- Substitute the values of w_x, w_y, and θ into the equations.
- Solve the equations simultaneously to find the values of T_x and T_y.
The tension in the cable (Tension) is equal to the magnitude of the tension vector, which can be calculated using the Pythagorean theorem: Tension = sqrt(T_x^2 + T_y^2).
The horizontal and vertical components of the force exerted by the boom on the pivot can be obtained from the values of T_x and T_y.
Remember to convert the angles from degrees to radians when using trigonometric functions.