If 2.4 J of work is needed to stretch a spring from 10 cm to 14 cm and another 4 J is needed to stretch it from 14 cm to 18 cm, what is the natural length (in cm) of the spring? cm
Kinematics
A machine is a device that accepts energy in some available form and utilizes it to do a type of work. Energy, work, or power has to be transferred from one mechanical part to another to run a machine. While the transfer of energy between two machine parts, those two parts experience a relative motion with each other. Studying such relative motions is termed kinematics.
Kinetic Energy and Work-Energy Theorem
In physics, work is the product of the net force in direction of the displacement and the magnitude of this displacement or it can also be defined as the energy transfer of an object when it is moved for a distance due to the forces acting on it in the direction of displacement and perpendicular to the displacement which is called the normal force. Energy is the capacity of any object doing work. The SI unit of work is joule and energy is Joule. This principle follows the second law of Newton's law of motion where the net force causes the acceleration of an object. The force of gravity which is downward force and the normal force acting on an object which is perpendicular to the object are equal in magnitude but opposite to the direction, so while determining the net force, these two components cancel out. The net force is the horizontal component of the force and in our explanation, we consider everything as frictionless surface since friction should also be calculated while called the work-energy component of the object. The two most basics of energy classification are potential energy and kinetic energy. There are various kinds of kinetic energy like chemical, mechanical, thermal, nuclear, electrical, radiant energy, and so on. The work is done when there is a change in energy and it mainly depends on the application of force and movement of the object. Let us say how much work is needed to lift a 5kg ball 5m high. Work is mathematically represented as Force ×Displacement. So it will be 5kg times the gravitational constant on earth and the distance moved by the object. Wnet=Fnet times Displacement.
![### Problem Statement
**If 2.4 J of work is needed to stretch a spring from 10 cm to 14 cm and another 4 J is needed to stretch it from 14 cm to 18 cm, what is the natural length (in cm) of the spring?**
\[ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \text{cm} \]
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### Explanation
In this problem, we are dealing with a spring and the work required to stretch it over certain distances. The information given can be used to find the natural (unstretched) length of the spring.
To solve this:
1. **Understand Hooke's Law:**
\[ F = kx \]
where \( F \) is the force applied, \( k \) is the spring constant, and \( x \) is the displacement from its natural length.
2. **Relate Work Done with Spring Stretching:**
The work \( W \) done to stretch a spring is given by:
\[ W = \frac{1}{2} k x^2 \]
3. **Use Provided Information**:
- 2.4 J to stretch from 10 cm to 14 cm.
- 4 J to stretch from 14 cm to 18 cm.
4. **Form Equations**:
- Work done from 10 cm to 14 cm:
\[ W_1 = \frac{1}{2} k (14 - L)^2 - \frac{1}{2} k (10 - L)^2 = 2.4 \, \text{J} \]
- Work done from 14 cm to 18 cm:
\[ W_2 = \frac{1}{2} k (18 - L)^2 - \frac{1}{2} k (14 - L)^2 = 4 \, \text{J} \]
5. **Solve the Equations**:
We solve these equations simultaneously to determine the natural length \( L \) of the spring.
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### Instructions for Solution
1. Form the equations using the given data.
2. Substitute the values and simplify the equations to find the unknown variables.
3. Solve the simultaneous equations to determine \( L \), the natural length of the spring.
### Conclusion
By solving the provided equations, you](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9fe49cee-acc8-403b-8d12-20ec3514f583%2Fe16e3de2-efaa-4c4b-a19e-c22a6486d1ea%2F2e0pwkm_processed.jpeg&w=3840&q=75)
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