Which of the following statements is incorrect? (a) The stress in a body is force per unit area (b) Hooke’s law holds good up to the breaking point (c) Stress is directly proportional to strain within elastic limit (d) The ratio of linear stress to linear strain is called Young’s modulus
Answers
Answer:
When studying springs and elasticity, the 17ᵗʰ century physicist Robert Hooke noticed that the stress vs strain curve for many materials has a linear region. Within certain limits, the force required to stretch an elastic object such as a metal spring is directly proportional to the extension of the spring. This is known as Hooke's law and commonly written:
\boxed{F=-kx}
F=−kx
start box, F, equals, minus, k, x, end box
Where FFF is the force, xxx is the length of extension/compression and kkk is a constant of proportionality known as the spring constant which is usually given in \mathrm{N/m}N/mN, slash, m.
Though we have not explicitly established the direction of the force here, the negative sign is customarily added. This is to signify that the restoring force due to the spring is in the opposite direction to the force which caused the displacement. Pulling down on a spring will cause an extension of the spring downward, which will in turn result in an upward force due to the spring.
It is always important to make sure that the direction of the restoring force is specified consistently when approaching mechanics problems involving elasticity. For simple problems we can often interpret the extension xxx as a 1-dimensional vector; in this case the resulting force will also be a 1-dimensional vector and the negative sign in Hooke’s law will give the correct direction of the force.
When calculating xxx, it is important to remember that the spring itself will also have some nominal length L_0L
0
L, start subscript, 0, end subscript. The total length LLL of a spring under extension is equal to the nominal length plus the extension, L=L_0 + xL=L
0
+xL, equals, L, start subscript, 0, end subscript, plus, x. For a spring under compression, it would be L=L_0-xL=L
0
−xL, equals, L, start subscript, 0, end subscript, minus, x.
Exercise 1: A 75 kg person stands on a compression spring with spring constant 5000~\mathrm{N/m}5000 N/m5000, space, N, slash, m and nominal length 0.25~\mathrm{m}0.25 m0, point, 25, space, m. What is the total length of the loaded spring? [Solution]
Exercise 2a: You are designing a mount for moving a 1 kg camera smoothly over a vertical distance of 50 mm. The design calls for the camera to slide on a pair of rails, with a spring supporting the camera and pulling it up against the tip of an adjustment screw as shown in Figure 1. The nominal length of the spring is L_0=50~\mathrm{mm}L
0
=50 mmL, start subscript, 0, end subscript, equals, 50, space, m, m.
Explanation:
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