Two bars have the same moment of inertia around their horizontal centroidal axes. The hollow bar has an outside diameter of 1.275 in. and a wall thickness of 0.255 in. Determine the diameter of the solid bar.

For a beam with the cross-section shown (w = 15.0 in. and h = 22.0 in.), find the distance from the bottom surface of the beam to the centroid.

A concentrated load of P = 30 kN is applied to the upper end of a 3-m-long pipe as shown. The outside diameter of the pipe is D = 330 mm and the inside diameter is d = 300 mm. Determine the magnitude of the maximum vertical shear stress in the pipe.

A 14-ft-long simply supported timber beam carries a P = 16-kip concentrated load at midspan as shown. The cross-sectional dimensions of the timber are also shown. At section a-a, determine the magnitude of the shear stress in the beam at point H.

Two identical beams are loaded with w and 2w. Which beam will have the shear force with the smallest magnitude?

The internal shear force V at a certain section of an aluminum beam is 33 kN. The beam’s centroid is located 54.40 mm above the bottom surface of the beam, and the moment of inertia is Iz = 398,300 mm4. Determine the shear stress at point H, which is located 30 mm above the bottom surface of the beam.

Two cylindrical beams each support a shear force of 10.7 N. The outside diameter of the hollow beam is 44 mm, and its wall thickness is 4 mm. Determine the diameter of the solid beam that would create the same value of Q in both beams.

The figure shows gradient descent on a convex cost function. The parameter θ\thetaθ starts near the bottom of the curve, but the updates repeatedly jump from one side of the valley to the other, moving far away from the minimum. What is the best interpretation of this behavior?

A student trains a linear regression model using gradient descent. One feature is house size measured from 500 to 5,000 square feet, while another feature is number of bedrooms measured from 1 to 6. The student does not scale the features, and gradient descent takes a long time to converge. Why is feature scaling important in this situation?

What is the relationship between the following two compounds? Screenshot 2026-06-15 at 1.05.27 PM.png