Understanding Work Done by a Force

In physics, the work done by a constant force \(\overrightarrow F\) on an object that moves through a displacement \(\widehat d\) is defined as the dot product (or scalar product) of the force vector and the displacement vector. This is a fundamental concept in mechanics and relates force, displacement, and energy transfer. The formula for calculating the work done is:

\[W = \overrightarrow F \cdot \widehat d\]

The result of a dot product is a scalar quantity, which means work done is a scalar, having magnitude but no direction.

Identifying the Given Force and Displacement Vectors

The problem provides us with the force vector and the displacement vector through which the object moves.

• The applied force vector is given as \(\overrightarrow F = 2\widehat i - \widehat j - \widehat k\).
• The displacement vector is given as \(\widehat d = 3\widehat i + 2\widehat j - 5\widehat k\).

Here, \(\widehat i\), \(\widehat j\), and \(\widehat k\) are the unit vectors along the x, y, and z axes, respectively, in a Cartesian coordinate system.

Calculating the Work Done using the Dot Product

To find the work done, we need to calculate the dot product of the force vector \(\overrightarrow F\) and the displacement vector \(\widehat d\).

Let \(\overrightarrow F = F_x\widehat i + F_y\widehat j + F_z\widehat k\) and \(\widehat d = d_x\widehat i + d_y\widehat j + d_z\widehat k\). The dot product is calculated as:

\[\overrightarrow F \cdot \widehat d = F_x d_x + F_y d_y + F_z d_z\]

Substituting the components from our given force and displacement vectors:

• \(F_x = 2\), \(F_y = -1\), \(F_z = -1\)
• \(d_x = 3\), \(d_y = 2\), \(d_z = -5\)

Now, we perform the calculation for the work done:

\[W = (2)(3) + (-1)(2) + (-1)(-5)\] \[W = 6 - 2 + 5\] \[W = 4 + 5\] \[W = 9\]

Therefore, the work done in moving the object along the given displacement vector by the applied force vector is 9 units. This is a standard physics calculation involving vectors. Understanding the concept of work done as a scalar product is crucial.

Final Result for Work Done

Based on the calculation of the dot product of the force vector and the displacement vector, the work done is 9 units. This result represents the energy transferred to the object by the force over the displacement.