A square steel plate is 10.0 cm on a side and 0.500 cm thick. (a) Find the shear strain that results if a force of magnitude 9.0×10^5 N is applied to each of the four sides, parallel to the side. (b) Find the displacement x (in centimeters).

Verified step by step guidance

1. Identify the given values: side length of the square plate (L) = 10.0 cm, thickness of the plate (t) = 0.500 cm, and the force applied (F) = 9.0×10^5 N.
2. Understand that shear strain (\( \gamma \)) is defined as the ratio of the displacement (x) to the original dimension perpendicular to the force, in this case, the thickness of the plate. The formula for shear strain is \( \gamma = \frac{x}{t} \).
3. Calculate the shear stress (\( \tau \)) using the formula \( \tau = \frac{F}{A} \), where A is the area over which the force is applied. Since the force is applied parallel to the sides, A equals the side length times the thickness (A = L \times t).
4. Use Hooke's Law for shear, which relates shear stress and shear strain through the shear modulus (G) of the material. The formula is \( \tau = G \times \gamma \). You will need to look up the shear modulus (G) for steel.
5. Substitute the expression for shear strain from step 2 into Hooke's Law for shear to find the displacement x. Rearrange the formula to solve for x: \( x = \frac{\tau \times t}{G} \).

Key Concepts

Shear Strain

Shear strain is a measure of how much a material deforms in response to an applied shear force. It is defined as the change in the angle between two lines originally at right angles, typically expressed as the ratio of the displacement to the height of the material. In this case, the shear strain can be calculated using the formula γ = Δx / h, where Δx is the displacement and h is the thickness of the plate.

Shear Stress

Shear stress is the force per unit area acting parallel to the surface of a material. It is calculated using the formula τ = F / A, where F is the applied force and A is the area over which the force is distributed. In the context of the steel plate, understanding shear stress is crucial for determining how the applied force affects the material's deformation and ultimately leads to shear strain.

Displacement

Displacement in this context refers to the amount of movement that occurs in the material due to the applied shear force. It can be calculated from the shear strain and the thickness of the material. The relationship between shear strain and displacement is essential for solving part (b) of the question, as it allows us to quantify how much the plate shifts under the applied load.