Physics 15 Torque (5 of 27) Tension=? Supporting a Gate

Michel van Biezen
13 Jan 201614:21
EducationalLearning
32 Likes 10 Comments

TLDRThe video script presents a detailed physics problem involving the concept of torque to find the tension in a cable suspending a gate and the forces at its hinges. The gate, with a mass of 400 Newtons and dimensions of one meter by two meters, is suspended at a 30-degree angle by a cable. The challenge is to calculate the tension in the cable and the vertical and horizontal components of force at the hinges, labeled A and B. The solution involves summing torques around a pivot point, using trigonometric relationships to find perpendicular distances, and applying equilibrium conditions. The process requires finding the perpendicular distance from the line of action of forces to the pivot point, which is key to solving for the tension. The video concludes with the calculation of the tension in the cable as 214 Newtons and the horizontal force at hinge B as 186 Newtons, illustrating a methodical approach to problem-solving in physics.

Takeaways
  • 📐 The problem involves a gate suspended by a cable at a 30-degree angle to the horizontal, with two hinges A and B, and a mass causing a weight of 400 Newtons.
  • ⚖️ The horizontal force at hinge A is zero, meaning there is no stress at A in the horizontal direction, only vertical.
  • 🔍 The goal is to find the tension in the cable and the three components of the forces: the vertical component at A and the horizontal and vertical components at B.
  • 🔩 Torques are summed around pivot point B to find the tension in the cable without needing to know the forces at B in the X and Y directions.
  • 📏 The perpendicular distance (D) from the line of action of the tension force to the pivot point is found using trigonometry based on the gate's dimensions and the angle of the cable.
  • 🧮 The tension in the cable (T) is calculated to be 214 Newtons by equating the torques caused by the gate's weight and the tension force.
  • 🔄 To find the force at B in the X direction (F_Bx), the pivot point is moved to A, and torques are summed around A, considering the tension and weight of the gate.
  • 🔢 The force at B in the X direction is found to be 186 Newtons after calculating the perpendicular distance (d2) from the line of action of force to the pivot point.
  • ⚙️ The sum of the forces in the Y direction equals zero, which is used to find the combined force at A and B in the Y direction, which is 93 Newtons.
  • 💡 The problem-solving approach involves using trigonometry to find distances and angles, and then applying these to calculate the forces and tensions involved.
  • 📉 A mistake in calculating the sine of 60 degrees was corrected during the process, emphasizing the importance of accuracy in trigonometric calculations.
Q & A

  • What is the main concept used to solve the problem presented in the lecture?

    -The main concept used to solve the problem is the concept of torque.

  • What are the dimensions of the gate mentioned in the lecture?

    -The gate has a height of one meter and a length of two meters.

  • What is the weight of the gate in Newtons?

    -The weight of the gate is 400 Newtons.

  • What is the angle of the cable with the horizontal?

    -The angle of the cable with the horizontal is 30 degrees.

  • Why is the horizontal force at hinge A equal to zero?

    -The tension in the cable is adjusted so that the horizontal force at hinge A is equal to zero, indicating no stress at hinge A in the horizontal direction.

  • What is the method used to find the tension in the cable?

    -The method used to find the tension in the cable is by summing up the torques about a pivot point, in this case, pivot point B.

  • How is the perpendicular distance (D) from the line of action of the force to the pivot point found?

    -The perpendicular distance (D) is found by first determining the distance Y using the tangent of 30 degrees and then using it to calculate the height H of the post. D is then found using the sine of the 60-degree angle and the height H.

  • What is the calculated tension in the cable?

    -The calculated tension in the cable is 214 Newtons.

  • How is the force at B in the X direction determined?

    -The force at B in the X direction is determined by summing up the torques about hinge A and solving for the force, which is found to be a positive 186 Newtons.

  • What is the sum of the vertical forces at A and B?

    -The sum of the vertical forces at A and B must be equal to 93 Newtons, as determined by the balance of forces in the Y direction.

  • What is the approach to find the vertical components of the forces at A and B?

    -The approach is to sum up all the forces in the Y direction and set the equation to zero, which gives the sum of the vertical forces at A and B.

  • What is the significance of the sine and cosine functions in this problem?

    -The sine and cosine functions are used to find the perpendicular and adjacent sides of the right triangles formed by the gate, cable, and hinges, which are essential for calculating the distances and forces involved.

Outlines
00:00
😀 Introduction to Torque Problem

This paragraph introduces the problem of finding the tension in a cable suspending a gate with two hinges. The gate has a mass of 400 Newtons and dimensions of 1m x 2m. The tension is adjusted so there is no horizontal force at hinge A. The goal is to find the tension in the cable and the three components of forces at the hinges. The approach is to sum up the torques about a pivot point, chosen as hinge B, and then solve for the unknowns.

05:01
🧐 Calculating Tension in the Cable

This paragraph focuses on finding the tension in the cable. The key steps are: 1) Summing torques about hinge B, which must equal zero. 2) Calculating the perpendicular distance D from the line of action of the cable tension to the pivot point. 3) Using the right triangle formed by the gate's height, the horizontal distance Y, and the vertical distance H to find D = 1.87m. 4) Solving for the tension T = (mg * a) / D = 214 N. The challenge is finding the perpendicular distance D from the line of action to the pivot point.

10:02
🔍 Finding Forces at Hinge B

This paragraph aims to find the forces at hinge B. The approach is to sum up the torques about hinge A, which must equal zero. The key steps are: 1) Setting up the equation with the tension force causing a positive torque, the weight of the gate causing a negative torque, and the horizontal force at B causing a positive or negative torque depending on its direction. 2) Solving for the horizontal force Bx = W - Td2 = 186 N, confirming the assumed direction of Bx is correct. 3) Summing the vertical forces to find the combined force at A and B in the Y direction equals 93 N. The challenge is solving for the two unknown vertical forces at A and B in the same line of action.

Mindmap
Keywords
💡Torque
Torque is a measure of the rotational force that can cause an object to rotate about an axis. In the video, it is used to analyze the forces acting on a gate suspended by a cable, with the goal of finding the tension in the cable and the forces at the hinges. The concept of torque is central to the problem-solving process, as it helps determine the balance of forces that keep the gate stable.
💡Gate
The gate in the video is a physical structure that is being analyzed for the forces acting upon it. It has a mass causing a weight of 400 Newtons, is one meter tall, and two meters long. The gate's dimensions and weight are crucial in calculating the forces and torques involved in the problem.
💡Cable
The cable is a rope or chain used to suspend the gate at an angle to the horizontal. The tension in the cable is a key unknown in the problem, and the video aims to find this value. The cable's angle of 30 degrees to the horizontal is significant in determining the forces and torques acting on the gate.
💡Hinges
Hinges A and B are the points where the gate is connected and rotate. The video discusses the horizontal force at hinge A being zero and the existence of forces at hinge B in both vertical and horizontal directions. Understanding the forces at the hinges is essential for analyzing the gate's stability.
💡Tension
Tension refers to the force exerted by a cable or rope when it is pulled tight. In the context of the video, finding the tension in the cable is one of the main objectives. The tension in the cable is directly related to the torques created by the weight of the gate and the forces at the hinges.
💡Force Components
Force components are the individual parts of a force vector that act in specific directions, typically horizontal (X) and vertical (Y). The video breaks down the forces at the hinges into horizontal and vertical components to solve for the unknown forces. These components are crucial for calculating torques and understanding the gate's equilibrium.
💡Pivot Point
A pivot point is a fixed point around which a body rotates. In the video, the pivot point is chosen at hinge B to simplify the calculation of torques. The selection of the pivot point is strategic and helps in determining the forces and tensions without directly knowing the forces at hinge B.
💡Summation of Torques
The summation of torques is the process of adding up all the torques acting on a system. In the video, it is stated that the sum of the torques about the pivot point must equal zero for the system to be in equilibrium. This principle is used to set up equations that can be solved for the unknowns.
💡Right Triangle
A right triangle is a geometric figure with one angle being 90 degrees. The video uses the properties of right triangles to find the perpendicular distances necessary for calculating torques. Specifically, it uses the tangent function and the sine function to find the lengths of the sides of the triangles formed by the gate and the cable.
💡Trigonometric Functions
Trigonometric functions, such as sine and tangent, are used to relate the angles and side lengths of triangles. In the video, these functions are essential for finding the distances from the lines of action of forces to the pivot point, which are needed to calculate the torques.
💡Equilibrium
Equilibrium refers to a state where all forces and torques acting on a body are balanced, resulting in no acceleration. The video's problem-solving approach is based on achieving equilibrium, where the sum of the torques equals zero, and the forces at the hinges are such that they counterbalance the weight of the gate.
Highlights

The lecture introduces a problem-solving approach using the concept of torque.

A gate is suspended by a cable at a 30-degree angle, with specific mass and dimensions provided.

The horizontal force at hinge A is zero, indicating no stress in the horizontal direction.

The problem aims to find the tension in the cable and the components of forces at hinges A and B.

Torques are summed around pivot point B to eliminate the need to know forces at B.

The weight of the gate contributes to a negative torque due to its downward pull.

Tension in the cable is expected to cause a positive counterclockwise torque.

The perpendicular distance from the line of action of the force to the pivot point is crucial for torque calculations.

The distance D is found using trigonometric relationships in a right triangle.

The tension in the cable is calculated to be 214 Newtons using the formula T = mg / D.

Finding the distance D involves breaking down the problem into finding Y and then H, the hypotenuse.

The force at B in the X direction is determined by summing torques around hinge A.

The direction of the force at B in the X direction is confirmed to be correct through calculation.

The sum of the forces at B in the Y direction and at A in the Y direction equals 93 Newtons.

The tension component in the Y direction is calculated using the sine of the 30-degree angle.

The problem-solving process involves a combination of torque calculations and trigonometric functions.

The lecture concludes by emphasizing the importance of understanding the line of action and perpendicular distances in torque problems.

Transcripts

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PhysicsTorqueOnline LectureEngineeringProblem SolvingForce CalculationSuspension MechanicsHinge AnalysisMathematical ApproachEducational ContentMechanics of Materials