How Much Work Is Done If A 4 Kg Object Is Raised 2.3 Meters At An Acceleration Of 4.5 M/s 2 4.5 \, \text{m/s}^2 4.5 M/s 2 ?

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Introduction

Work is a fundamental concept in physics that describes the transfer of energy from one object to another through a force applied over a distance. In this article, we will delve into the calculation of work done when raising an object, specifically a 4 kg object, 2.3 meters at an acceleration of 4.5 m/s24.5 \, \text{m/s}^2. We will explore the underlying physics, derive the necessary equations, and provide a step-by-step solution to determine the work done.

Understanding Work and Energy

Work is defined as the product of the force applied to an object and the distance over which the force is applied. Mathematically, work (W) is expressed as:

W = F * d

where F is the force applied and d is the distance over which the force is applied.

However, when dealing with objects under the influence of gravity or other external forces, the concept of potential energy becomes relevant. Potential energy (PE) is the energy an object possesses due to its position or configuration. For an object raised to a height h, the potential energy is given by:

PE = m * g * h

where m is the mass of the object, g is the acceleration due to gravity (approximately 9.8 m/s29.8 \, \text{m/s}^2), and h is the height to which the object is raised.

Calculating Work Done

To calculate the work done in raising the 4 kg object 2.3 meters, we need to consider the force required to overcome the weight of the object. The weight (W) of the object is given by:

W = m * g

Substituting the values, we get:

W = 4 kg * 9.8 m/s29.8 \, \text{m/s}^2 = 39.2 N39.2 \, \text{N}

The force required to raise the object is equal to its weight, which is 39.2 N39.2 \, \text{N}.

Using the Work-Energy Theorem

The work-energy theorem states that the net work done on an object is equal to the change in its kinetic energy (KE). Mathematically, this is expressed as:

W = ΔKE

Since the object is raised to a height, its kinetic energy remains constant, and the work done is equal to the change in potential energy.

Deriving the Equation for Work Done

To derive the equation for work done, we need to consider the force applied to the object as a function of time. The force (F) is given by:

F = m * a

where a is the acceleration of the object.

Substituting the values, we get:

F = 4 kg * 4.5 m/s24.5 \, \text{m/s}^2 = 18 N18 \, \text{N}

The work done (W) is then given by:

W = ∫F * dx

where dx is the infinitesimal distance over which the force is applied.

Substituting the expression for F, we get:

W = ∫(m * a) * dx

Evaluating the integral, we get:

W = m * a * h

Substituting the values, we get:

W = 4 kg * 4.5 m/s24.5 \, \text{m/s}^2 * 2.3 m = 41.4 J41.4 \, \text{J}

Conclusion

In conclusion, the work done in raising a 4 kg object 2.3 meters at an acceleration of 4.5 m/s24.5 \, \text{m/s}^2 is 41.4 J41.4 \, \text{J}. This calculation demonstrates the application of the work-energy theorem and the use of the work done equation to determine the energy transferred to an object.

References

  • [1] Halliday, D., Resnick, R., & Walker, J. (2013). Fundamentals of Physics. John Wiley & Sons.
  • [2] Serway, R. A., & Jewett, J. W. (2018). Physics for Scientists and Engineers. Cengage Learning.

Additional Resources

Q&A: Frequently Asked Questions

Q: What is work in physics?

A: Work is a measure of the energy transferred to an object by a force applied over a distance. It is calculated as the product of the force applied and the distance over which the force is applied.

Q: How is work related to energy?

A: Work is a form of energy transfer. When a force is applied to an object, it can transfer energy to the object, causing it to move or change its state. The work done on an object is equal to the change in its kinetic energy.

Q: What is the difference between work and energy?

A: Work is the transfer of energy from one object to another through a force applied over a distance. Energy, on the other hand, is the ability to do work. There are various forms of energy, including kinetic energy, potential energy, and thermal energy.

Q: How is work calculated?

A: Work is calculated using the formula:

W = F * d

where W is the work done, F is the force applied, and d is the distance over which the force is applied.

Q: What is the work-energy theorem?

A: The work-energy theorem states that the net work done on an object is equal to the change in its kinetic energy. Mathematically, this is expressed as:

W = ΔKE

Q: How is work done in raising an object calculated?

A: To calculate the work done in raising an object, we need to consider the force required to overcome the weight of the object. The weight (W) of the object is given by:

W = m * g

where m is the mass of the object and g is the acceleration due to gravity.

Q: What is the significance of the work done in raising an object?

A: The work done in raising an object is a measure of the energy transferred to the object. It is an important concept in physics, as it helps us understand how energy is transferred and transformed from one form to another.

Q: Can work be negative?

A: Yes, work can be negative. When a force is applied in the opposite direction to the motion of an object, the work done is negative. This is because the force is opposing the motion, and therefore, the energy transferred to the object is negative.

Q: What is the unit of work?

A: The unit of work is the joule (J). It is a derived unit, equal to the energy transferred when a force of 1 newton is applied over a distance of 1 meter.

Q: Can work be zero?

A: Yes, work can be zero. When a force is applied perpendicular to the motion of an object, the work done is zero. This is because the force is not doing any work on the object, and therefore, the energy transferred is zero.

Q: What is the relationship between work and potential energy?

A: The work done in raising an object is equal to the change in its potential energy. Mathematically, this is expressed as:

W = ΔPE

where W is the work done and PE is the potential energy.

Q: Can work be transferred from one object to another?

A: Yes, work can be transferred from one object to another through a force applied over a distance. This is an important concept in physics, as it helps us understand how energy is transferred and transformed from one form to another.

Q: What is the significance of the work done in a system?

A: The work done in a system is a measure of the energy transferred to the system. It is an important concept in physics, as it helps us understand how energy is transferred and transformed from one form to another.

References

  • [1] Halliday, D., Resnick, R., & Walker, J. (2013). Fundamentals of Physics. John Wiley & Sons.
  • [2] Serway, R. A., & Jewett, J. W. (2018). Physics for Scientists and Engineers. Cengage Learning.

Additional Resources