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A vertical load $\mathbf{W}$ is applied to the linkage at $B$. The constant of the spring is $k$, and the spring is unstretched when $A B$ and $B C$ are horizontal. Neglecting the weight of the linkage, derive an equation in $u, W, l,$ and $k$ that must be satisfied when the linkage is in equilibrium.

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Physics 101 Mechanics

Chapter 10

Method of Virtual Work

Work

Potential Energy

Cornell University

Rutgers, The State University of New Jersey

Simon Fraser University

McMaster University

Lectures

02:08

In physics, work is the transfer of energy by a force acting through a distance. The "work" of a force F on an object that it pushes is defined as the product of the force and the distance through which it moves the object. For example, if a force of 10 newtons (N) acts through a distance of 2 meters (m), then doing 10 joules (J) of work on that object requires exerting a force of 10 N for 2 m. Work is a scalar quantity, meaning that it can be described by a single number-for example, if a force of 3 newtons acts through a distance of 2 meters, then the work done is 6 joules. Work is due to a force acting on a point that is stationary-that is, a point where the force is applied does not move. By Newton's third law, the force of the reaction is equal and opposite to the force of the action, so the point where the force is applied does work on the person applying the force. In the example above, the force of the person pushing the block is 3 N. The force of the block on the person is also 3 N. The difference between the two forces is the work done on the block by the person, which can be calculated as the force of the block times the distance through which it moves, or 3 N × 2 m = 6 J.

03:23

In physics, mechanical energy is the sum of the kinetic and potential energies of a system.

06:45

A vertical load $\mathbf{P…

07:36

A load $\mathrm{W}$ with a…

09:49

A load $\mathbf{W}$ with a…

for this problem on the topic off equilibrium of rigid bodies were told that a vertical load is applied at the end of Iran, as shown in diagram were given the concept of the spring to be K and were given the angle deter. When the spring is on stretched, we want to calculate or express rather the angle theta corresponding to the equilibrium position in terms of p k l. We didn't want to determine the value of theater if we had given a value off off P and equilibrium. So here we have a free body diagram off our rod. We could break up the reaction foresee into its X and Y components. The load PX vertically downwards and the tea is the force along the spring. Now we can see the triangle, ABC and Socialists. And so we have a be equal to to a D, But we choose yeah, to l sign Deter Over to and C D is equal to al times the co sign off data over to now for the elongation of the spring. We'll do the following so the elongation of the spring will call X and X is equal to to l sign Utah over to minus to l sign 30 degrees. Yeah. Now we have an expression for the 11 gauge, not the spring X, And we can write an expression for tea. So we know the force acting on the spring. My hopes law is K X. So we could write this as by substituting expression for X from above as to K L. Into sign off data over to minus sign of 30 degrees at equilibrium, which is a half. And now we can use this expression later on. So if we take anti clockwise moments about point See, we know that the some of these moments and equilibrium must equal to zero. So taking the some of the moments, this is the forcing the spring t times the distance al times the co sign off Dita over to my interest in the opposite direction. You load P and Al Sign. Peter was equal to zero. So we can We can bring in our expression for the attention or the force on the spring tea into this equation. And we can rewrite the second term as well. So we can write this as to que el into sign Peter over to minus half into l co sign data over to minus pl. And we can expand the outside Lita as a double egg. No expansion. So that's to sign data over to co sign data over to remember. Sign of two data is to cite data courtside data. No, this is equal to zero. So this implies that either because I'm off Peter over to is equal to zero, uh, to into que el minus p? Yeah. Times the sign off detail over to minus K. L is equal to zero. So with two solutions are that TRIVOLI data must equal to 180 degrees or the sign off Peter over to must equal. Uh huh. Que el divided by que el minus the load p and so we can write expression for theater. So theater needs to be two times the art sign off a half. Que el divided by que el minus p. So here we have an expression for the the angle data in terms of PK and L. And that's our ancestral plot. A now for part B. We want to find this angle, Dita. If we had given a value for p. So we're told that equilibrium at an equilibrium P is able to 1/4 k l. And so, if we go back to our expression for Dita or rather for sign Peter, we have signed data over to is equal to half KL over the over four que el, which is simplifies to two thirds. Yeah. So this means that Dita over to is able to 41.8 degrees, which means that we can find Tita simply in this condition to be 83 0.6 degrees.

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