A drawbridge with a weight of 5000 N is supported by a hinge and rope and is in static equilibrium. The length of the drawbridge is 5 m and the rope is attached to the draw bridge a distance of 4 m from the hinge. What is the tension force rope provides?
Can you show me steps to do this problem?
Thanks!
Certainly! To find the tension force that the rope provides, we can use the principle of torque. Here are the steps to solve the problem:
Step 1: Draw a diagram: Sketch the drawbridge and label the given information. In this case, draw the drawbridge with a weight of 5000 N, a length of 5 m, and mark the location where the rope is attached at a distance of 4 m from the hinge.
Step 2: Identify the forces: Identify the forces acting on the drawbridge. In this case, we have two forces: the weight of the drawbridge (5000 N) acting downward and the tension force from the rope acting upward.
Step 3: Define the pivot point: Choose a point as the pivot around which torques will be calculated. In this case, the hinge is the pivot point.
Step 4: Set up the torque equation: The torque equation is given by the product of the force and the lever arm. Torque = force x lever arm. In this problem, since the drawbridge is in static equilibrium, the sum of the torques must be zero.
Step 5: Calculate the torque: Calculate the torque due to the weight of the drawbridge. Torque = force x lever arm. The force (weight) is 5000 N and the lever arm is the distance between the hinge and the center of mass of the drawbridge, which is half its length, i.e., 5 m / 2 = 2.5 m.
Step 6: Calculate the lever arm of the tension force: The lever arm for the tension force is the perpendicular distance between the hinge and the point of attachment of the rope. In this case, it is given as 4 m.
Step 7: Set up the torque equation: The sum of the torques must be zero, so we equate the two torques: Torque due to the weight = Torque due to the tension force.
Step 8: Solve for the tension force: Rearrange the equation to solve for the tension force. Tension force = (Force due to weight x Lever arm of weight) / Lever arm of tension force. Plug in the values obtained from steps 5 and 6 to find the tension force.
By following these steps, you should be able to calculate the tension force provided by the rope.
Sure! Here are the steps to solve this problem:
Step 1: Draw a diagram of the drawbridge showing all the forces acting on it. Label the weight of the drawbridge as 5000 N, the length of the drawbridge as 5 m, and the distance from the hinge to the point where the rope is attached as 4 m.
Step 2: Identify all the forces acting on the drawbridge. In this case, there are two forces: the weight of the drawbridge acting downwards, and the tension force of the rope acting upwards.
Step 3: Apply the principle of torque. Torque is the product of a force and the perpendicular distance from the force to the pivot point. In this case, the hinge serves as the pivot point. The torque due to the weight of the drawbridge is equal to the weight multiplied by the perpendicular distance from the weight to the hinge. Since the weight is acting downward, the perpendicular distance is the distance from the hinge to the center of mass of the drawbridge, which is half of the length of the drawbridge (5 m / 2 = 2.5 m).
Step 4: Set up the equation for the torque due to the weight of the drawbridge. Torque = Force × Distance. In this case, Torque = Weight × Perpendicular Distance. Thus, Torque = 5000 N × 2.5 m = 12500 N·m.
Step 5: Set up the equation for the torque due to the tension force of the rope. The torque due to the tension force of the rope is equal to the tension force multiplied by the perpendicular distance from the tension force to the hinge. Since the tension force is acting upwards, the perpendicular distance is the distance from the hinge to the point where the rope is attached (4 m).
Step 6: Set the torques equal to each other. Since the drawbridge is in static equilibrium, the torques due to all the forces acting on it must balance each other. Therefore, Torque due to weight = Torque due to tension force.
Step 7: Solve for the tension force. Using the equation from Step 6, we can solve for the tension force by dividing both sides of the equation by the perpendicular distance. Tension Force = Torque due to weight / Perpendicular distance. Substituting the values we have, we get: Tension Force = 12500 N·m / 4 m = 3125 N.
Therefore, the tension force provided by the rope is 3125 N.