Lecture Notes: Towing a Car on an Inclined Plane

Problem Statement

• A car is being towed up an inclined plane.
• Incline Angle: 20 degrees
• Applied Force: 5000 Newton
• Car Mass: 1000 kg
• Movement: Constant speed (no acceleration)
• Negligible Friction
• Maximum Force Ramp Can Withstand: 9500 Newton

Objective

• Determine if the ramp will collapse or withstand the force when the car is towed.

Steps to Solve

Step 1: Identify Forces

• Free Body Diagram required to visualize forces.

• Coordinate system:

  • X-axis along the incline.
  • Y-axis perpendicular to the incline.

• Forces Involved:

  1. Applied Force (F)
  2. Normal Force (N)
  3. Weight of the Car (mg)

Step 2: Resolve Forces

• Weight Component Resolution:
  • Break weight into components parallel and perpendicular to the incline.
  • X-direction: Weight component = mg * sin(theta)
  • Y-direction: Weight component = mg * cos(theta)

Step 3: Analyze Forces in X-Direction

• Forces: Applied force (F) and weight component (mg * sin(theta))

• Equation: The net force = 0 (since no acceleration)

  • F - mg * sin(theta) = 0
  • F = mg * sin(theta)

• Calculation:

  • F = 1000 * 9.8 * sin(20)
  • Result: F = 3352 Newton

• Conclusion:

  • The towing force (3352 Newton) is less than the rope's capacity (5000 Newton).
  • Therefore, the rope will not break.

Step 4: Analyze Forces in Y-Direction

• Forces: Normal force (N) and weight component (mg * cos(theta))

• Equation: Net force = 0 (no vertical movement)

  • N - mg * cos(theta) = 0
  • N = mg * cos(theta)

• Calculation:

  • N = 1000 * 9.8 * cos(20)
  • Result: N = 9209 Newton

• Conclusion:

  • The normal force (9209 Newton) is less than the ramp's capacity (9500 Newton).
  • Therefore, the ramp will not collapse.

Additional Considerations

• If friction is present, it must be included in calculations as an additional force opposing movement up the incline.

Summary

• The system is safe: the rope will not break, and the ramp will not collapse under the given conditions.

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