- #1

Luke Tan

- 29

- 2

- Homework Statement
- A steel plate of thickness h has the shape of a square whose side equals l, with h << l. The plate is rigidy fixed to a vertical axle OO which is rotated with a constant angular acceleration ##\beta##. Find the deflection ##\lambda##, assuming the sagging to be small

- Relevant Equations
- $$EI\frac{d^2y}{dx^2}=N(x)$$

My struggle here comes from finding the bending moment ##N(x)##. My working is as follows.

We want to find the bending moment on an element a distance ##x## away from the axis of rotation. To do so, let us consider the bending moment due to the force on an element ##\xi>x## away from the axis of rotation. The perpendicular force on this element is ##\xi\beta dm##. Hence, the total bending moment on this element would be

$$N=\int_x^l(\xi-x)(\xi\beta \mu d\xi)$$

Where ##\mu=\rho l h## is the mass per unit length. However, the solutions manual has this expression

(Solutions to I.E. Irodov's Problems in General Physics, Volume 1, Second Edition, Singh)

Where the integrand is ##\xi^2##, rather than ##\xi(\xi-x)##. I initially thought that the solutions manual was wrong, but it seems like sources all over the internet have ##\xi^2## under the integral sign. Can someone explain to me what I did wrong?

Thanks!

We want to find the bending moment on an element a distance ##x## away from the axis of rotation. To do so, let us consider the bending moment due to the force on an element ##\xi>x## away from the axis of rotation. The perpendicular force on this element is ##\xi\beta dm##. Hence, the total bending moment on this element would be

$$N=\int_x^l(\xi-x)(\xi\beta \mu d\xi)$$

Where ##\mu=\rho l h## is the mass per unit length. However, the solutions manual has this expression

(Solutions to I.E. Irodov's Problems in General Physics, Volume 1, Second Edition, Singh)

Where the integrand is ##\xi^2##, rather than ##\xi(\xi-x)##. I initially thought that the solutions manual was wrong, but it seems like sources all over the internet have ##\xi^2## under the integral sign. Can someone explain to me what I did wrong?

Thanks!