A force \(F =\left(5+3 y^2\right)\) acts on a particle in the \(y\)-direction, where \(F\) is in newton and \(y\) is in meter The work done by the force during a displacement from \(y=2 m\) to \(y=5 m\) is___ \(J\).
A force \(F =\left(5+3 y^2\right)\) acts on a particle in the \(y\)-direction, where \(F\) is in newton and \(y\) is in meter The work done by the force during a displacement from \(y=2 m\) to \(y=5 m\) is___ \(J\).
Show Hint
Work done by a variable force is calculated by integrating the force with respect to displacement. Ensure the force and displacement are in the same direction.
Correct Answer: 132
Solution and Explanation
Step 1: Recall the Formula for Work Done
The work done (\(W\)) by a variable force \(F(y)\) over a displacement from \(y_1\) to \(y_2\) is given by the integral:
\[ W = \int_{y_1}^{y_2} F(y) \, dy \]
Step 2: Substitute the Given Force and Limits
In this case, \(F(y) = 5 + 3y^2\), \(y_1 = 2 \, \text{m}\), and \(y_2 = 5 \, \text{m}\). So,
\[ W = \int_{2}^{5} (5 + 3y^2) \, dy \]
Step 3: Evaluate the Integral
\[ W = \left[ 5y + \frac{3y^3}{3} \right]_2^5 = \left[ 5y + y^3 \right]_2^5 \] \[ W = (5(5) + 5^3) - (5(2) + 2^3) \] \[ W = (25 + 125) - (10 + 8) \] \[ W = 150 - 18 = 132 \, \text{J} \]
Conclusion: The work done by the force is \(132 \, \text{J}\).
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Concepts Used:
Work, Energy and Power
Work:
- Work is correlated to force and the displacement over which it acts. When an object is replaced parallel to the force's line of action, it is thought to be doing work. It is a force-driven action that includes movement in the force's direction.
- The work done by the force is described to be the product of the elements of the force in the direction of the displacement and the magnitude of this displacement.
Energy:
- A body's energy is its potential to do tasks. Anything that has the capability to work is said to have energy. The unit of energy is the same as the unit of work, i.e., the Joule.
- There are two types of mechanical energy such as; Kinetic and potential energy.
Read More: Work and Energy
Power:
- Power is the rate at which energy is transferred, conveyed, or converted or the rate of doing work. Technologically, it is the amount of work done per unit of time. The SI unit of power is Watt (W) which is joules per second (J/s). Sometimes the power of motor vehicles and other machines is demonstrated in terms of Horsepower (hp), which is roughly equal to 745.7 watts.
- Power is a scalar quantity, which gives us a quantity or amount of energy consumed per unit of time but with no manifestation of direction.