A horizontal cantilever beam of circular crosssection, length 1.0 m and flexural rigidity EI = 200 N.m² is subjected to an applied moment M_A= 1.0 N-m at the free end as shown in the figure. The magnitude of the vertical deflection of the free end is _____________mm (round off to one decimal place).

The correct answer is 2.5.

Step-by-Step Explanation:

We are given a horizontal cantilever beam of circular cross-section fixed at the left end and free at the right end. The parameters provided are:
Length of the beam, $L=1.0\text{m}$
Flexural rigidity of the beam, $EI=200{\text{N·m}}^2$
Applied moment at the free end, $M_A=1.0\text{N·m}$

The image confirms that the beam of length 1.0 m is fixed on the left and has a counterclockwise concentrated moment $M_A$ acting at the free end on the right.

For a cantilever beam subjected to a concentrated bending moment $M$ at its free end, the standard formula for the maximum deflection (which occurs at the free end) is given by:
$y_{\text{free end}}=\frac{M_A{L}^2}{2EI}$

Substituting the given values into the deflection formula:
$y_{\text{free end}}=\frac{1.0\times {1.0}^2}{2\times 200}$
$y_{\text{free end}}=\frac{1}{400}\text{m}$
$y_{\text{free end}}=0.0025\text{m}$

To convert this deflection from meters (m) to millimeters (mm), we multiply by 1000:
$y_{\text{free end}}=0.0025\times 1000\text{mm}=2.5\text{mm}$

Thus, the magnitude of the vertical deflection of the free end is 2.5 mm.