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Reaction forces of a hook wrench

Task

A hook wrench is attached to a stationary shaft. Which reaction forces arise in A and B under the assumption that the contact in B is frictionless?

Hook wrench on shaft
Hook wrench on the shaft

Solution

The following video is in german language. Please scroll down for the written solution.

Reaktionskräfte bestimmen - Technische Mechanik 1, Übung 15
Reaction forces on the shaft
Reaction forces on the shaft
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The reaction forces on the shaft are shown in the sketch of the solution. The hook wrench touches the shaft at points A and B. On a spatial shaft a line contact would occur.

The equilibrium of forces in the x-direction is

\[\tag{1} \sum F_x = 0 = F_B -- F_{Ax} \]

\[\tag{2} F_{Ax} = F_B \]

The equilibrium of forces in the y-direction is

\[\tag{3} \sum F_y = 0 = F_{Ay} -- F \]

\[\tag{4} F_{Ay} = F \]

The equilibrium of moments is established around the axis of rotation of the shaft. The moment applied by the force F is referred to as M F , so that the moment equilibrium reads

\[\tag{5} \sum M = 0 = -- M_F + F_A \cdot \frac{d}{2} \]

\[\tag{6} F_A \cdot \frac{d}{2} = F \cdot l \]

\[\tag{7} F_A = \frac{2 \cdot F \cdot l}{d} \]

The still unknown horizontal forces are calculated using the Pythagorean theorem:

\[\tag{8} F_A^2 = F_{Ax}^2 + F_{Ay}^2 \]

\[\tag{9} \left( \frac{2 \cdot F \cdot l}{d} \right)^2 = F_{Ax}^2 + F^2 \]

\[\tag{10} F_{Ax} = \sqrt{\left( \frac{2 \cdot F \cdot l}{d} \right)^2 -- F^2} \]