A vertical spring of force constant 100 N / m is attached with a hanging mass of 10 kg. Now an

A vertical spring of force constant 100 N / m is attached...

A vertical spring of force constant $100 \mathrm{~N} / \mathrm{m}$ is attached with a hanging mass of $10 \mathrm{~kg}$. Now an external force is applied on the mass so that the spring is stretched by additional $2 \mathrm{~m}$. The work done by the force $F$ is $\left(\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^{2}\right)$ (A) $200 \mathrm{~J}$ (B) $400 \mathrm{~J}$ (C) $450 \mathrm{~J}$ (D) $600 \mathrm{~J}$

A vertical spring of force constant $100 \mathrm{~N} / \mathrm{m}$ is attached with a hanging mass of $10 \mathrm{~kg}$. Now an external force is applied on the mass so that the spring is stretched by additional $2 \mathrm{~m}$. The work done by the force $F$ is $\left(\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^{2}\right)$
(A) $200 \mathrm{~J}$
(B) $400 \mathrm{~J}$
(C) $450 \mathrm{~J}$
(D) $600 \mathrm{~J}$

Key Concepts

Work-Energy Principle

The work-energy principle states that the work done by external forces on a system results in a change in the system's energy. In the context of a mass-spring system, when an external force stretches the spring, the work done is divided between increasing the elastic potential energy stored in the spring and changing the gravitational potential energy of the mass. This principle is essential for analyzing how forces applied to a system translate into changes in energy.

Hooke's Law

Hooke's Law describes the linear relationship between the force exerted by a spring and its displacement from the equilibrium position, expressed as F = -kx, where k is the spring constant and x is the displacement. This principle is fundamental in understanding how springs behave under loads and how the restoring force is proportional to the stretch or compression of the spring.

Elastic Potential Energy

Elastic potential energy is the energy stored in an elastic object, such as a spring, when it is deformed from its equilibrium state. For a spring, this energy is given by the formula 1/2 k x^2, where k is the spring constant and x is the displacement. This concept is key in analyzing energy conservation and energy transfer in systems involving elastic objects.

Gravitational Potential Energy

Gravitational potential energy is the energy possessed by an object due to its position in a gravitational field. It is calculated using the formula mgh, where m is the mass of the object, g is the acceleration due to gravity, and h is the height above a reference level. When objects are lifted or lowered in a gravitational field, changes in gravitational potential energy must be considered in the overall energy balance of the system.

Transcript

00:01 In this question we have given a vertical spring whose spring constant is 100 newton per meter and a block is attached to it it is told that an external force is applied on this mass and the spring is stretched by an additional of 2 meter so we have to find the work done by this force so let we have given diagram this is our spring mass system here now let the extension of this spring due to the weight is this is the extension that is equal to the weight is the extension that is equal to the spring to x1 and let due to the force it have an additional extension that is equal to this is our x2 so we can say that this is our neutral position so this is x is equal to zero here so we can say that x is that is x1 is extension due to weight of block which so we can say that at equilibrium this weight is equal to this is m g into kx at this point the force weight acting vertically down with mg and this is the spring force which is k x1 so we can say that from here x1 is equal to this is mg by k which is equal to 10 into 10 upon 100 that is 1 meter so this is the extension due to the weight now it is told that it has additional extension of 2 meter so we can say that x2 is equal to this is x1 plus 2 which is 3 meter now we know that work done by applied force plus work done by this gravity force is equal to this is change in potential energy of the spring that is half of k x2 square minus x1 square so from here we can say that this is worked done by applied force plus this is m g into this is is x2 minus x1 is equal to half of k this is x2 square minus x1 square now put all the data in this equation so we can say that work done by applied force is equal to this is half into 100 into this is 3 square minus 1 square minus this is 10 into 10 into 2.

02:24 We can say this will come out to be 200 jule.

02:28 So this is the work done by applied force...

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