# Derivation of equation for mass on pulley and displacement

• TrippingBilly
In summary, the conversation discusses the derivation of an equation for the mass on a pulley and its displacement. The system is in equilibrium and consists of three masses: A, B, and M. The equation for the vertical displacement of the center mass, denoted as h, is given as h= ML / sqrt(16m^2 - 4M^2). The conversation also includes equations for the sum of forces in the x and y directions and the tensions T1-T5 represented as mg and Mg. However, the relationship between T5 and T2/T3 is unclear.
TrippingBilly
http://filesaur.us/files/1858/pulley/

Derivation of equation for mass on pulley and displacement

Sorry that the picture stinks, but its all I got. The system is in equilibrium.The counter mass on the left is mass A and the mass on the right is mass B, both of mass m. The center mass of mass M is denoted as B. The length of the system is denoted as L. h stands for the vertical displacement of the center mass. The equation is..

h= ML / sqrt(16m^2 - 4M^2)

I wrote the equations for the sum of the forces and my teacher told me I could derive it from those but I can't get any further than what I have.
Forces in x direction = T(sub c)cos(theta) - T(sub a)cos(theta) =0 and
Forces in y direction = T(sub c)sin(theta) + T(sub a)sin(theta) - T(sub b) = 0

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You don't have the tensions T described in the drawing.

Also -- can you give us expressions for the tensions in terms of m, M, L, h?

http://filesaur.us/files/1912/pulley/
T1 = T2 = T3 = T4 = mg
T5=Mg

I don't know how to bring L or h into this.

Last edited by a moderator:
In what relation is T5 with T2 and T3?

## 1. What is the equation for mass on pulley and displacement?

The equation for mass on pulley and displacement is derived from the principles of mechanical advantage and conservation of energy. It is expressed as m1d1 = m2d2, where m1 is the mass on one side of the pulley, d1 is the displacement of that mass, m2 is the mass on the other side of the pulley, and d2 is the displacement of that mass.

## 2. How is this equation derived?

The equation is derived by considering the forces acting on the masses and the work done by those forces. By setting the work done by the heavier mass equal to the work done by the lighter mass, we can derive the equation m1d1 = m2d2.

## 3. What are some practical applications of this equation?

This equation can be used in various situations, such as determining the weight of an unknown mass by using a known mass and measuring the displacement, or calculating the displacement needed to lift a certain mass using a pulley system. It is also commonly used in physics and engineering experiments involving pulleys.

## 4. Are there any limitations to this equation?

Yes, this equation assumes an idealized situation where there is no friction or other external forces acting on the system. In real-life scenarios, there will always be some level of friction and other factors that may affect the accuracy of the equation.

## 5. Can this equation be applied to pulley systems with more than two masses?

Yes, this equation can be extended to pulley systems with more than two masses by incorporating additional terms for each mass and displacement involved. However, the concept of mechanical advantage and conservation of energy still apply in these situations.

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