Q13-31. The tractor is used to lift the 150-kg load B with the 24-m-long rope, boom, and pulley system. If the tractor travels to the right at a constant speed of 4 m/s, determine the tension in the rope when m. When , .

Background

Topic: Dynamics of Systems Involving Pulleys and Ropes

This question tests your understanding of how to analyze the forces and kinematics in a system involving a moving tractor, a pulley, and a suspended load. You need to relate the motion of the tractor to the motion of the load and determine the tension in the rope.

Key Terms and Formulas

• Tension (): The force transmitted through the rope.

• Kinematic Constraint: The total length of the rope is constant, so the positions of the tractor and the load are related.

• Newton's Second Law:

• Geometry of the System: Use the Pythagorean theorem to relate (horizontal distance of tractor) and (vertical position of load):

• Constant Speed: Acceleration of the load is zero.

Step-by-Step Guidance

1. Write the constraint equation for the rope length, relating and using the geometry shown in the diagram.

2. Since the tractor moves at a constant speed, determine the velocity of the load by differentiating the constraint equation with respect to time.

3. Apply Newton's Second Law to the load in the vertical direction. Since the load moves at constant velocity, its acceleration is zero.

4. Set up the force balance: , where is the mass of the load and is the acceleration due to gravity.

Try solving on your own before revealing the answer!

Q13-32. The tractor is used to lift the 150-kg load B with the 24-m-long rope, boom, and pulley system. If the tractor travels to the right with an acceleration of 3 m/s2 and has a velocity of 4 m/s at the instant m, determine the tension in the rope at this instant. When , .

Background

Topic: Dynamics of Systems Involving Pulleys and Ropes (with Acceleration)

This question extends the previous one by introducing acceleration of the tractor, which means the load will also experience acceleration. You need to relate the acceleration of the tractor to the acceleration of the load and then determine the tension in the rope.

Key Terms and Formulas

• Tension (): The force in the rope supporting the load.

• Kinematic Constraint: The rope length is constant, so the positions, velocities, and accelerations of the tractor and load are related.

• Newton's Second Law:

• Geometry of the System: Use the Pythagorean theorem as before to relate and .

• Chain Rule for Acceleration: Differentiate the constraint equation twice to relate the accelerations of and .

Step-by-Step Guidance

1. Write the constraint equation for the rope length, relating and using the geometry shown in the diagram.

2. Differentiate the constraint equation twice with respect to time to relate the acceleration of the tractor () to the acceleration of the load ().

3. Calculate the acceleration of the load using the given acceleration of the tractor and the relationship from the constraint equation.

4. Apply Newton's Second Law to the load in the vertical direction: .

Try solving on your own before revealing the answer!