Question

Calculate by direct integration the moment of inertia for a thin rod of mass M and length L about an axis located distance d from one end. Confirm that your answer agrees with Table 12.2 when d=0 and when d = L/2.

Accepted Answer

Step 1: Define the moment of inertia formula for a continuous mass distribution. The moment of inertia is given by $$ I = \int r^2 \, dm $$, where $$ r  is $$the distance from the axis of rotation to the mass element $$ dm . $$Step 2: Express $$ dm  in $$terms of the linear mass density $$ \lambda . $$For a thin rod, $$ \lambda = \frac{M}{L} $$, where $$ M  is $$the total mass and $$ L  is $$the length of the rod. Thus, $$ dm = \lambda \, dx = \frac{M}{L} \, dx $$, where $$ dx  is $$the infinitesimal length element. Step 3: Set up the integral for the moment of inertia. The distance $$ r $$ from the axis of rotation to the mass element depends on the position $$ x $$ along the rod. If the axis is located at a distance $$ d $$ from one end, then $$ r = x - d . $$Substitute $$ r $$ and $$ dm $$ into the formula: $$ I = \int_{d}^{d+L} (x - d)^2 \frac{M}{L} \, dx . $$Step 4: Simplify the integral. Expand $$ (x - d)^2  to$$ $$ x^2 - 2dx + d^2 $$, and distribute $$ \frac{M}{L} $$: $$ I = \frac{M}{L} \int_{d}^{d+L} (x^2 - 2dx + d^2) \, dx . $$Break this into three separate integrals: $$ I = \frac{M}{L} \left[ \int_{d}^{d+L} x^2 \, dx - 2d \int_{d}^{d+L} x \, dx + d^2 \int_{d}^{d+L} 1 \, dx \right] . $$Step 5: Evaluate each integral. Use the standard rules for integration: $$ \int x^n \, dx = \frac{x^{n+1}}{n+1} $$, $$ \int x \, dx = \frac{x^2}{2} $$, and $$ \int 1 \, dx = x . $$Substitute the limits of integration $$ d $$ and $$ d+L $$ into each term. After simplifying, confirm the result for $$ d = 0 $$ and $$ d = \frac{L}{2}  to $$verify agreement with Table 12.2.

Calculate by direct integration the moment of inertia for a thin rod of mass M and length L about an axis located distance d from one end. Confirm that your answer agrees with Table 12.2 when d=0 and when d = L/2.

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Key Concepts

Moment of Inertia

Direct Integration

Axis of Rotation