A spring with spring constant k=200 N / m is suspended vertically with its upper end fixed to t

A spring with spring constant k=200 N / m is suspended...

A spring with spring constant $k=200 \mathrm{~N} / \mathrm{m}$ is suspended vertically with its upper end fixed to the ceiling and its lower end at position $y=0 .$ A block of weight $20 \mathrm{~N}$ is attached to the lower end, held still for a moment, and then released. What are (a) the kinetic energy $K,$ (b) the change (from the initial value) in the gravitational potential energy $\Delta U_{g},$ and $(\mathrm{c})$ the change in the elastic potential energy $\Delta U_{e}$ of the spring-block system when the block is at $y=-5.0 \mathrm{~cm} ?$ What are $(\mathrm{d}) K,(\mathrm{e}) \Delta U_{g},$ and $(\mathrm{f}) \Delta U_{e}$ when $y=-10 \mathrm{~cm},(\mathrm{~g}) K,(\mathrm{~h}) \Delta U_{g},$ and (i) $\Delta U_{e}$ when $y=-15 \mathrm{~cm}$ and $(\mathrm{j}) K,(\mathrm{k}) \Delta U_{g},$ and (l) $\Delta U_{e}$ when $y=-20 \mathrm{~cm} ?$

A spring with spring constant $k=200 \mathrm{~N} / \mathrm{m}$ is suspended vertically with its upper end fixed to the ceiling and its lower end at position $y=0 .$ A block of weight $20 \mathrm{~N}$ is attached to the lower end, held still for a moment, and then released. What are (a) the kinetic energy $K,$ (b) the change (from the initial value) in the gravitational potential energy $\Delta U_{g},$ and $(\mathrm{c})$ the change in the elastic potential energy $\Delta U_{e}$ of the spring-block system when the block is at $y=-5.0 \mathrm{~cm} ?$ What are $(\mathrm{d}) K,(\mathrm{e}) \Delta U_{g},$ and $(\mathrm{f}) \Delta U_{e}$ when $y=-10 \mathrm{~cm},(\mathrm{~g}) K,(\mathrm{~h}) \Delta U_{g},$ and
(i) $\Delta U_{e}$ when $y=-15 \mathrm{~cm}$
and $(\mathrm{j}) K,(\mathrm{k}) \Delta U_{g},$ and
(l) $\Delta U_{e}$ when $y=-20 \mathrm{~cm} ?$

Key Concepts

Kinetic Energy

Kinetic energy is the energy associated with the motion of an object. It is determined by the mass of the object and the square of its velocity, illustrating how the energy increases as the speed of the object increases. This concept is crucial when assessing energy exchange in dynamic systems where moving masses are involved.

Gravitational Potential Energy

Gravitational potential energy is the energy possessed by an object due to its position in a gravitational field. It depends on the object's mass, the gravitational acceleration, and the height of the object relative to a defined reference point. This energy form is significant in problems involving vertical displacement and energy transformations between potential and kinetic energy.

Elastic Potential Energy

Elastic potential energy is the energy stored in objects that can be deformed, such as springs. When a spring is compressed or stretched from its equilibrium position, energy is stored in the spring, which can later be converted into kinetic energy or other forms. This concept is foundational for understanding systems involving elastic forces and Hooke’s law.

Conservation of Mechanical Energy

The principle of conservation of mechanical energy states that in an isolated system, where only conservative forces are doing work, the total mechanical energy (the sum of kinetic energy, gravitational potential energy, and elastic potential energy) remains constant. This concept is pivotal in predicting the behavior of a system as energy is transformed from one form to another without any losses.

Transcript

00:01 So we have a question here in which there is a ceiling and a spring is attached to the ceiling.

00:08 Now we have put a block of mass 20 newtons or i should i should say the weight of 20 newton.

00:15 The block has a weight of 20 newtons.

00:23 Now the spring constant of the spring is 200 newton per meter.

00:32 Now this is this is a reference level.

00:35 Y is equals to 0.

00:37 At this position the spring is not elongated.

00:40 We have just put the block.

00:43 What happens afterwards is that this block will elongate the spring and there will be four positions in which we'll be talking about in this question.

01:08 At this position y equals minus 0 .05 meter at this position y equals minus 0 .10 meter at this position y equals minus 0 .10 meter at this position y equals minus 0 .15 meters at this position y equals minus 0 .15 meters and at this position y equals minus 0 .20 meters now at y is equals to 0 or the reference level the total energy of the body the total energy of the body equals 0 the total energy of the system i should say equals 0 and the first subpart what we need to calculate is and before before talking about the first sub part let's talk about the formula or the conservation of energy formula according to this conservation of energy the initial the total energy of the of the system at y equals to 0 equals kinetic energy plus gravitation potential energy plus the elastic potential energy at various positions so after rearranging we got k equals minus ug minus uv now in the first subpart we are asked that we need to calculate the kinetic energy at position y equals to minus 0 .05 meters.

02:52 In this first part we need to calculate the kinetic energy at minus 0 .05 meters.

02:58 So equals minus u g minus ue.

03:02 So that is equal to minus w into y, minus of minus w into y because the potential energy is negative minus half into k into y squared.

03:17 So the value of minus y into y is minus 1 and the value of elastic potential energy is 0 .25.

03:33 This will be in jules.

03:36 So the final value will be 1 minus 0 .25 joules that is equal to 0 .75 joules...

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