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Chapter 10: Problem 25

The upper surface of a cube of gelatin, \(5.0 \mathrm{cm}\) on a side, is displaced \(0.64 \mathrm{cm}\) by a tangential force. If the shear modulus of the gelatin is \(940 \mathrm{Pa},\) what is the magnitude of the tangential force?

Short Answer

Expert verified
Answer: The magnitude of the tangential force is 0.3008 N.

Step by step solution

01

Write down the shear stress formula

To find the tangential force, we need to calculate the shear stress using the formula: Shear stress = \(\frac{F}{A}\) Where \(F\) is the tangential force and \(A\) is the area of the surface the force is applied on.
02

Write down the definition of shear modulus

Shear modulus, denoted by \(G\), is defined as the ratio of shear stress (\(\tau\)) to shear strain (\(\gamma\)), that is: \(G = \frac{\tau}{\gamma}\)
03

Calculate the shear strain

Shear strain is defined as the ratio of the displacement of the upper surface to the side length of the cube. In our case, the displacement is \(0.64 \mathrm{cm}\) and the side length is \(5.0 \mathrm{cm}\). Therefore, the shear strain is: \(\gamma = \frac{0.64\,\mathrm{cm}}{5.0\,\mathrm{cm}} = 0.128\)
04

Calculate the shear stress with the shear modulus

Using the shear modulus given, we can find the shear stress: \(\tau = G\gamma = 940\,\mathrm{Pa} \times 0.128 = 120.32\,\mathrm{Pa}\)
05

Calculate the area where the force is applied

The force is applied on the upper surface of the cube. The area of this surface is: \(A = (5.0\,\mathrm{cm})^2 = 25\,\mathrm{cm^2} = 2.5 \times 10^{-3}\,\mathrm{m^2}\) (converted to square meters)
06

Calculate the tangential force

Now we can use the shear stress formula to find the tangential force: \(F = \tau A = 120.32\,\mathrm{Pa} \times 2.5 \times 10^{-3}\,\mathrm{m^2} = 0.3008\,\mathrm{N}\) So, the magnitude of the tangential force is \(0.3008\,\mathrm{N}\).

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