The moment of inertia of a uniform cylinder about its perpendicular bisector is given by the formula I = (M/12) * (3R^2 + l^2), where M is the mass of the cylinder, R is the radius, and l is the length.

To find the ratio l/R that minimizes the moment of inertia, we need to take the derivative of I with respect to the ratio l/R and set it equal to zero.

Let's denote the ratio l/R as x. Then, the formula for the moment of inertia becomes I = (M/12) * (3 + x^2).

Taking the derivative of I with respect to x, we get dI/dx = (M/6) * x.

Setting this equal to zero, we find that x = 0. This means that the moment of inertia is minimized when the length l is zero.

However, this doesn't make sense in the context of the problem, because we're dealing with a cylinder, which by definition has a non-zero length.

Therefore, the ratio l/R that minimizes the moment of inertia of a uniform cylinder about its perpendicular bisector doesn't exist.

Question

The moment of inertia of a uniform cylinder about its perpendicular bisector is given by the formula I = (M/12) * (3R^2 + l^2), where M is the mass of the cylinder, R is the radius, and l is the length.

To find the ratio l/R that minimizes the moment of inertia, we need to take the derivative of I with respect to the ratio l/R and set it equal to zero.

Let's denote the ratio l/R as x. Then, the formula for the moment of inertia becomes I = (M/12) * (3 + x^2).

Taking the derivative of I with respect to x, we get dI/dx = (M/6) * x.

Setting this equal to zero, we find that x = 0. This means that the moment of inertia is minimized when the length l is zero.

However, this doesn't make sense in the context of the problem, because we're dealing with a cylinder, which by definition has a non-zero length.

Therefore, the ratio l/R that minimizes the moment of inertia of a uniform cylinder about its perpendicular bisector doesn't exist.

Knowee AI · Accepted Answer

The moment of inertia of a uniform cylinder about its perpendicular bisector is given by the formula I = (M/12) * (3R^2 + l^2), where M is the mass of the cylinder, R is the radius, and l is the length.

To find the ratio l/R that minimizes the moment of inertia, we need to take the derivative of I with respect to the ratio l/R and set it equal to zero.

Let's denote the ratio l/R as x. Then, the formula for the moment of inertia becomes I = (M/12) * (3 + x^2).

Taking the derivative of I with respect to x, we get dI/dx = (M/6) * x.

Setting this equal to zero, we find that x = 0. This means that the moment of inertia is minimized when the length l is zero.

However, this doesn't make sense in the context of the problem, because we're dealing with a cylinder, which by definition has a non-zero length.

Therefore, the ratio l/R that minimizes the moment of inertia of a uniform cylinder about its perpendicular bisector doesn't exist.

The moment of inertia of a uniform cylinder of length l and radius R about its perpendicular bisector is I. What is the ratio l/R such that the moment of inertia is minimum ?

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