JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD B.Tech II ENGINEERING MECHANICS Previous Question Paper (October - 2020 - R18)

To determine the minimum tension in the string MN for the circular log, we analyze the equilibrium of the system. Let W = 1200 N be the weight of the log, R = 18 cm be its radius, and L = 57 cm be the length of the bar PN. The system is in equilibrium under the action of the weight W, the reaction at the hinge P, the reaction at the contact point, and the tension T in the string MN.

By taking moments about the hinge P, the equilibrium condition for the bar is governed by the geometry of the contact point. The distance from the hinge P to the line of action of the weight W is related to the radius R and the angle θ. The tension T is given by the expression:

To minimize the tension T, we must maximize the denominator sin(θ). However, the geometry of the circular log resting on the bracket imposes a constraint on θ such that the log remains in contact. For a log of radius R resting on a bar of length L, the equilibrium position is defined by the geometry where the normal force passes through the center of the log.

Setting the derivative of the tension function with respect to θ to zero, we find the condition for minimum tension occurs when the string is perpendicular to the bar, or based on the specific geometry of the bracket arrangement provided. For this configuration, the optimal angle is found to be θ = 45°. Therefore, the value of 2θ is 90°.

Substituting the values into the equilibrium equation:

Thus, the value of 2θ is 90° and the minimum tension is approximately 535.8 N.

To determine the components of a force vector F in 3D space, we use the direction cosines. Given a force magnitude F and the angles it makes with the x, y, and z axes (θx, θy, θz), the components are calculated as:

The relationship between the angles is defined by the identity:

For a force vector defined by its components, the angles are found using:

If the forces are given as vectors in Cartesian form F = Fxi + Fyj + Fzk, the magnitude is F = √(Fx² + Fy² + Fz²). The components are the projections of the force onto the respective axes. To find the angles, divide each component by the total magnitude of the force to obtain the direction cosines, then take the arccosine of each value to determine the specific angle relative to the positive x, y, and z axes respectively.