A long, uniform rope with a mass of 0.135 kg per meter lies...

A long, uniform rope with a mass of 0.135 kg per meter lies on the ground. You grab one end of the rope and lift it at the constant rate of $1.13 \mathrm{m} / \mathrm{s}$. Calculate the upward force you must exert at the moment when the top end of the rope is $0.525 \mathrm{m}$ above the ground.

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Key Concepts

Mass per Unit Length

Mass per unit length, or linear density, is the measure of how much mass is contained in a unit length of an object. This concept is essential in problems involving ropes or strings, where the mass of any segment is determined by multiplying its length by the linear density.

Gravitational Force

Gravitational force is the force exerted by gravity on a mass, calculated by multiplying the mass by the acceleration due to gravity. In lifting problems, the force you must exert is at least equal to the gravitational force acting on the part of the object that is being lifted.

Constant Velocity Motion

Constant velocity motion means there is no acceleration, indicating that the net force on the object is zero. In a lifting scenario at constant speed, the applied force must exactly balance the gravitational force, ensuring that the object is lifted steadily without additional acceleration.

Lifting of a Continuous Object

When lifting a continuous object like a rope, the portion of the object that is off the ground increases gradually. The force required at any moment is determined by the weight of the lifted segment, which is found by multiplying the lifted length by the mass per unit length and the gravitational acceleration.

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