Springs, Hooke's Law, Work, and Energy

In summary, the formulas given include potential energy, stored strain energy, kinetic energy, work, and force. At the point in the vault where the pole is maximally deflected, Yelena Isinbayeva has a stored strain energy of 1151.11 N/m, a work done by the pole of 1151.11 N/m, a force applied to the pole of 1632.78 N/m, a kinetic energy in the Y direction of 651.52 N/m, and a gravitational kinetic energy of 1990.84 N/m. At the peak of her trajectory, the components of her mechanical energy will include potential energy, kinetic energy, and gravitational kinetic energy.
  • #1
x2022
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Homework Statement
The above image shows Yelena Isinbayeva (69.5 kg) at a point in her vault where the pole is maximally deflected (Δx = 1.41 m). The pole has a bending stiffness of 1158 N/m and behaves according to Hooke's Law. At this point in the vault, her vertical velocity is 4.33 m/s and she is 2.92 m above the ground.

If all of the stored stain energy is transferred from the pole to the vaulter, then what will be the peak height of her center of mass (relative to the ground) during the vault?
Relevant Equations
Stored Stain energy stored in pole = 1151.11
Work done by pole = 1151.11
Force applied to pole = 1632.78
Kinetic energy in Y = 651.52
Gravitational Kinetic Energy = 1990.84
The formulas we have been given include Potential energy=mgh, Stored strain energy=(1/2)K(change in X)^2, , Kinetic energy=(1/2)mV^2, Work=F(change in d), Force=K(change in X). Not sure how exactly to answer the question.
 
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  • #2
x2022 said:
Homework Statement: The above image shows Yelena Isinbayeva (69.5 kg) at a point in her vault where the pole is maximally deflected (Δx = 1.41 m). The pole has a bending stiffness of 1158 N/m and behaves according to Hooke's Law. At this point in the vault, her vertical velocity is 4.33 m/s and she is 2.92 m above the ground.

If all of the stored stain energy is transferred from the pole to the vaulter, then what will be the peak height of her center of mass (relative to the ground) during the vault?
Homework Equations: Stored Stain energy stored in pole = 1151.11
Work done by pole = 1151.11
Force applied to pole = 1632.78
Kinetic energy in Y = 651.52
Gravitational Kinetic Energy = 1990.84

The formulas we have been given include Potential energy=mgh, Stored strain energy=(1/2)K(change in X)^2, , Kinetic energy=(1/2)mV^2, Work=F(change in d), Force=K(change in X). Not sure how exactly to answer the question.
You have found all the components of her mechanical energy at the point in the (missing) diagram.
What will be the components of her mechanical energy at the peak of her trajectory?
 

What is Hooke's Law?

Hooke's Law states that the force exerted by a spring is directly proportional to the distance it is stretched or compressed from its equilibrium position. This relationship can be expressed mathematically as F = -kx, where F is the force, k is the spring constant, and x is the displacement from equilibrium.

How does a spring store and release energy?

When a spring is stretched or compressed, work is done on it, which causes potential energy to be stored in the spring. This potential energy is released when the spring returns to its equilibrium position, converting back into kinetic energy as the spring oscillates back and forth.

What factors affect the spring constant?

The spring constant, k, is affected by the material and dimensions of the spring. Materials with higher elasticity, such as steel, will have a higher spring constant. Additionally, a longer or thicker spring will also have a higher spring constant.

How is work calculated for a spring?

The work done on a spring can be calculated by multiplying the force applied to the spring by the distance it is stretched or compressed. This can be expressed as W = F * x, where W is the work, F is the force, and x is the displacement from equilibrium.

What is the relationship between work and energy in a spring?

Work and energy are closely related in a spring. The work done on a spring is equal to the change in potential energy of the spring. As the spring is stretched or compressed, work is done on it, increasing its potential energy. This potential energy is then released as the spring returns to equilibrium, converting back into kinetic energy.

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