A cylindrical aluminum rod, with an initial length of and radius , is clamped in place at one end and then stretched by a machine pulling parallel to its length at its other end. Assuming that the rod’s density (mass per unit volume) does not change, find the force magnitude that is required of the machine to decrease the radius to . (The yield strength is not exceeded.)
The magnitude of force applied on the cylindrical rod by machine is .
The initial length of the rod,
The initial radius of the rod,
The final radius of the rod,
You can use the concept of modulus of elasticity, and expression of density and volume of the rod. The formulas used are given below.
One end of the rod is clamped and the other end is stretched by the machine in such a way that the rod’s density remains the same after elongation of the rod.
The expression of the density of the rod is
The expression of the volume of the rod before and after elongation is:
Change in length of the rod due to the applied force by the machine is:
The expression of Young’s modulus of elasticity is:
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?
In Fig. 12-41, a climber with a weight of 533.8 N is held by a belay rope connected to her climbing harness and belay device; the force of the rope on her has a line of action through her center of mass. The indicated angles are and . If her feet are on the verge of sliding on the vertical wall, what is the coefficient of static friction between her climbing shoes and the wall?
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