Final answer:
The wood flexes perpendicular to its length by 0.12 mm due to the applied force and doesn't compress lengthwise significantly.
Step-by-step explanation:
To determine the amount of flex on the wood and the compression lengthwise, we can use the concept of torque and the stress-strain relationship.
(a) Flex perpendicular to its length:
To calculate the flex perpendicular to the length, we need to find the torque caused by the vertical force. Using the formula for torque, τ = r x F, we can find the torque as the product of the distance from the joint to the force and the magnitude of the force. This torque will cause the wood to flex perpendicularly to its length.
The torque is given by: τ = (2.00 cm)(6.00 N) = 12.00 N·cm.
Next, we can use the formula for the flexural rigidity of the wood, D = FL/3EI, where F is the applied force, L is the length of the wood, E is the Young's modulus, and I is the moment of inertia.
Since we are given the torque, we can use the relationship τ = FL/2, where F is the applied force, L is the length of the wood, and I is the moment of inertia, to find the flexural rigidity, D = τ/2.
Substituting the values, we get: D = (12.00 N·cm)/(2) = 6.00 N·cm.
Finally, we can use the formula for the deflection of a cantilever beam, δ = (F * L³)/(3 * E * I), to find the deflection. Substituting the values, we get: δ = (6.00 N * (60.00 mm)³)/(3 * (1 x 10⁹ N/m²) * (6.00 mm)⁴) = 0.12 mm.
(b) Compression lengthwise:
To calculate the compression lengthwise, we can use the formula for stress, σ = F/A, where F is the applied force and A is the cross-sectional area of the wood. The force causing the compression is the vertical force applied.
The area of the wood can be calculated using the formula for the area of a circle, A = πr², where r is the radius of the wood.
Substituting the values, we get A = π(3.00 mm)² = 28.27 mm².
Substituting the values, we get σ = (6.00 N)/(28.27 mm²) = 0.212 N/mm².
Finally, we can use the formula for the axial strain, ε = σ/E, where ε is the strain, σ is the stress, and E is the Young's modulus, to find the strain. Substituting the values, we get: ε = (0.212 N/mm²)/(1 x 10⁹ N/m²) = 2.12 x 10⁻⁷ mm/mm.
Since the strain is very small, we can assume that the wood doesn't compress lengthwise significantly. Therefore, the compression lengthwise is negligible.