What\'s the difference between stable and unstable equilibrium? What features of

Question

What's the difference between stable and unstable equilibrium? What features of a potential energy function do they correspond to? (b) You hang a 4 kg crate from the ceiling using a spring. The unstretched length of the spring is 10 cm. With the crate connected, the spring stretches out to a length of 15 cm. The system is at rest, with the crate hanging 2 meters above the ground. (b1) What is the stiffness (spring constant) of the spring? (b2) What is the total potential energy of the system? (b3) How much work does the force of gravity do on the crate if it gets disconnected and falls to the ground? Is the work positive or negative? (a) What's the difference between a scalar and a vector? Name two vector and two scalar quantities. Briefly describe their physical significance. (b) You are pushing a 10 kg crate up a ramp that tilted 30 degrees above the horizontal. The crate is moving with a constant velocity. You are applying 100 N of force to the crate. (b1) Identify the forces acting on the crate and draw diagram. (b2) what are the equations of motion for the crate? (b3) What type of friction is acting on the crate? What is the coefficient of friction? (b4) You increase the force of your push to 200 N. What is the crate's acceleration? What is the direction of its acceleration vector?

Explanation / Answer

(a)

For equilibrium condition two conditions must satisfy:

1. The sum of forces on a body must be zero.

2. Sum of all troques on the body about any point must be zero.

Stable equilibrium - a body in stable equilibrium, if diplaced slightly from it equilibrium position, tries to maintain its equilibrium.

Unstable equilibrium - a body in unstable equilibrium, if diplaced slightly from it equilibrium position, can't maintain its equilibrium. After a slight displacement from its equilibrium position it's equilibrium is permanently disturbed.

Potential energy function regarding stable equilibrium and unstable equilibrium :- The derivative of potential energy is the force on the system. Since, for an equilibrium, sum of forces must be zero, the derivative of the potenital energy must also be zero.

But, if the second derivative of the potential energy function is negative, the potential function is at a local maximum and this means that the system is in unstable equilibrium.

And, if the second derivative of the potential energy function is positive, the potential function is at a local minimum and this means that the system is in unstable equilibrium.

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(b)

(b1)

Let the spring force on the crate be F and let the extension in the spring be x,

x = 15 cm - 10 cm = 5 cm = 0.05 m

Then we have,

Mg = F = kx

or, k = Mg/x = (4 kg)(9.8 m/s2)/(0.05 m)

or, k = 784 N/m is the spring cosntant.

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(b2)

The potential energy of the spring is,

PEs = (1/2)kx2 = (0.5)(784 N/m)(0.05 m)2

or, PEs = 0.98 J

The potential energy of the earth-crate system, considering the gorund level to be at zero potential is,

PEg = Mgh = (4 kg)(9.8 m/s2)(2 m)

or, PEg = 78.4 J

So total potential energy of the system is,

PE = PEs + PEg = 0.98 J + 78.4 J

or, PE = 79.38 J

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(b3)

Work done by the gravity when the crate falls on the ground is,

W = mgh = (4 kg)(9.8 m/s2)(2 m) (here mg is the force of gravity and h is the displacement of the crate)

or, W = 78.4 J

The work done is positive as the force of gravity and displacement of the crate are both along the same direction.

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This concludes the answer. Check the answer and let me know if it's correct. If you need any more clarification or correction, feel free to ask.....

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