A variable force \(F=5x\) \(N\) acts on a body moving along x-axis. Find the work done by this force in displacing the body from \(x=2\) m to \(x=4\) m.
(K is constant)
(K is constant)
- \(\left(\frac{205}{2}K\right) J\)
- \(\left(\frac{105}{2}K\right) J\)
\(52K J\)
\(51K J\)
The Correct Option is B
Solution and Explanation
When force remains constant and motion occurs along a straight path, work is calculated as the product of force and distance traveled. However, when force varies within the distance traveled, calculus aids in resolving the problem by breaking it down into infinitesimal steps where force can be considered constant.
Consequently, the work accomplished by a variable force is expressed as,
\(W = ∫F dx\)
Here, the force is described by \(F = 5x\) N, and the displacement ranges from \(x = 2\) m to \(x = 4\) m.
Now the work performed,
\(W=∫Fdx\)
\(W=∫5x dx\)
\(W= 5∫_2^4 x dx\)
\(W = 5[\frac {x^2}{2}]_2^4\)
\(W=\frac 52[4^2-2^2]\)
\(W= 5\times 6\)
\(W=30\ J\)
So, the correct option is (B): \(\left(\frac{105}{2}K\right) J\)
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- Given below are two statements:
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Statement II: Any object moving from one location to another location can follow infinite number of paths. Therefore, the amount of work done by the object changes with the path it follows for a conservative force.
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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.