In the Figure, a lead brick rests horizontally on cylinders A and B. The areas of the top faces of the cylinders are related by AA=2AB; the Young’s moduli of the cylinders are related by EA=2EB. The cylinders had identical lengths before the brick was placed on them. What fraction of the brick’s mass is supported (a) by cylinder A and (b) by cylinder B? The horizontal distances between the center of mass of the brick and the centerlines of the cylinders are dA for cylinder A and dB for cylinder B. (c) What is the ratio dA/dB ?
The areas of faces of the cylinders are related by and Young moduli of the cylinders by
Using formula for Young’s modulus and torque, we can find the magnitude of the forces on the log from wire A and wire B and the ratio respectively.
Here, E is Young’s modulus, F is force, A is area, I is change in length, and L is original length.
Consider the formula for the torque:
data-custom-editor="chemistry" ….. (ii)
Here, data-custom-editor="chemistry" is torque, F is force, d is perpendicular distance.
From equation (i) for cylinder A and solve as:
Similarly, for cylinder B solve as:
data-custom-editor="chemistry" …… (iv)
The change in the length of both cylinders is the same. Therefore, equate equations (iii) and (iv) and simplify them further as,
Consider the equation as:
Solve further as:
The fraction of bricks mass supported by cylinder is 0.80
Consider the ratio:
The fraction of bricks mass supported by cylinder is 0.20
Applying torque equation about the center of mass is written as follows:
Substitute the values and solve as:
Therefore, the ratio is 0.25.
Question: Figure 12-29 shows a diver of weight 580 N standing at the end of a diving board with a length of L =4.5 m and negligible v mass. The board is fixed to two pedestals (supports) that are separated by distance d = 1 .5 m . Of the forces acting on the board, what are the (a) magnitude and (b) direction (up or down) of the force from the left pedestal and the (c) magnitude and (d) direction (up or down) of the force from the right pedestal? (e) Which pedestal (left or right) is being stretched, and (f) which pedestal is being compressed?
A crate, in the form of a cube with edge lengths of , contains a piece of machinery; the center of mass of the crate and its contents is located above the crate’s geometrical center. The crate rests on a ramp that makes an angle with the horizontal. As is increased from zero, an angle will be reached at which the crate will either tip over or start to slide down the ramp. If the coefficient of static friction between ramp and crate is (a) Does the crate tip or slide? And (b) at what angle does the crate tip or slide occur? (c) If , does the crate tip or slide? And (d) If , at what angle u does the crate tip or slide occur?
The system in Fig. 12-38 is in equilibrium. A concrete block of mass hangs from the end of the uniform strut of mass. A cable runs from the ground, over the top of the strut, and down to the block, holding the block in place. For angles and , find (a) the tension T in the cable and the (b) horizontal and (c) vertical components of the force on the strut from the hinge.
In Fig. 12-69, a package of mass m hangs from a short cord that is tied to the wall via cord 1 and to the ceiling via cord 2. Cord 1 is at angle with the horizontal; cord 2 is at angle. (a) For what value of is the tension in cord 2 minimized? (b) In terms of mg, what is the minimum tension in cord 2
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