Linear mass density and dangling rope.

In summary, The equation proposed to find the tension of a rope with mass M and length L, that is hanging from a branch and only affected by gravity, is F(r) = (Mg/L)(r), where r is the distance measured upward from the bottom of the rope.
  • #1
Timothy S
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A rope of mass M and length of L is dangling from a branch. The only force acting on it is gravity. The question is how do you find the tension of the rope at any position on the rope.

My proposed equation is this: F(r) = (Mg/L)(r) where r is any position on the rope. Is this equation correct? If it is incorrect what equation would be able to describe what I am asking?
 
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  • #2
Timothy S said:
A rope of mass M and length of L is dangling from a branch. The only force acting on it is gravity. The question is how do you find the tension of the rope at any position on the rope.

My proposed equation is this: F(r) = (Mg/L)(r) where r is any position on the rope. Is this equation correct? If it is incorrect what equation would be able to describe what I am asking?
It is correct if r is the distance measured upward from the bottom of the rope.

Chet
 

FAQ: Linear mass density and dangling rope.

1. What is linear mass density?

Linear mass density, also known as linear density, is a measure of the mass per unit length of a one-dimensional object. It is typically denoted by the symbol λ (lambda) and has units of kilograms per meter (kg/m).

2. How is linear mass density calculated?

Linear mass density is calculated by dividing the total mass of an object by its total length. For example, if a rope has a mass of 2 kilograms and a length of 4 meters, its linear mass density would be 0.5 kg/m (2 kg / 4 m = 0.5 kg/m).

3. How is linear mass density related to tension in a dangling rope?

Linear mass density is related to tension in a dangling rope through the equation T = λg, where T is the tension, λ is the linear mass density, and g is the acceleration due to gravity. This equation shows that as the linear mass density of a rope increases, so does the tension in the rope.

4. What is the difference between linear mass density and linear density?

Linear mass density and linear density are two terms that are often used interchangeably. However, linear density can also refer to the linear mass density of a two-dimensional object, such as a sheet of paper, whereas linear mass density specifically refers to the linear mass density of a one-dimensional object.

5. How is linear mass density affected by changes in length and mass?

Linear mass density is directly affected by changes in length and mass. If the length of an object increases, but the mass stays the same, the linear mass density will decrease. Similarly, if the mass increases while the length stays the same, the linear mass density will increase. This relationship can be expressed mathematically as λ = m / L, where λ is the linear mass density, m is the mass, and L is the length.

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