The drawing shows a 156-kg crate hanging from the end of a steel bar (S = 8.1 x 1010 N/m2). The length of the bar is 0.117 m, and its cross-sectional area is 7.44 x 10-4 m2. Neglect the weight of the bar itself and determine (a) the shear stress on the bar and (b) the vertical deflection Y of the right end of the bar.
I know that I have to use the equation to solve, but don't know how to start, If you could help explain the steps, I'll appreciate it.
got it,
a) the stress is
F/A= mg/A = 156(9.8)/7.44*10^-4= 2.05*10^6 N/m^2
B) F=S(delta y/ Lo)A,, solve for delta y
delta y= (F/A)*(Lo/S)
the value of F/A was found in part a, so substitute the answer in the second equation.
2.05*10^6(0.117/8.1*10^10)=2.96*10^-6m
Sure, I'd be happy to help you solve this problem with a touch of humor!
To find the shear stress on the bar, we can use the formula:
Shear stress = Force/Area
First, we need to find the force acting on the bar. Since the crate is hanging from the bar, the force is equal to the weight of the crate. So, the force is given by:
Force = mass x gravity
Force = 156 kg x 9.8 m/s^2 (approximating the acceleration due to gravity as 9.8 m/s^2)
Force ≈ 1528.8 N
Now that we have the force, we can calculate the shear stress by dividing the force by the cross-sectional area of the bar:
Shear stress = 1528.8 N / 7.44 x 10^-4 m^2
Shear stress ≈ 2.06 x 10^6 N/m^2
So, the shear stress on the bar is approximately 2.06 x 10^6 N/m^2. That's quite a tongue twister!
Now let's move on to find the vertical deflection of the right end of the bar. For this, we can use Hooke's Law, which states that the vertical deflection Y is equal to the force acting on the bar divided by the product of the bar's length and its material's Young's modulus.
In this case, the force acting on the bar is the weight of the crate, as we found earlier (1528.8 N). The length of the bar is given as 0.117 m, and the Young's modulus of the steel is 8.1 x 10^10 N/m^2.
Using Hooke's Law, we can calculate the vertical deflection as follows:
Y = Force / (Length x Young's modulus)
Y = 1528.8 N / (0.117 m x 8.1 x 10^10 N/m^2)
Y ≈ 1.93 x 10^-8 m
So, the vertical deflection of the right end of the bar is approximately 1.93 x 10^-8 m. That's quite a tiny deflection!
I hope these steps helped you understand how to solve the problem. Remember, when it comes to physics, it's important to stay grounded and keep the puns deflected!
To solve this problem, we can follow these steps:
Step 1: Determine the shear stress on the bar (a):
Since the crate is hanging from the end of the bar, the load from the crate creates a shear force on the bar. The shear stress (τ) is given by the formula:
τ = F / A
where F is the shear force and A is the cross-sectional area of the bar.
In this case, the shear force is equal to the weight of the crate, which is given by:
F = m * g
where m is the mass of the crate and g is the acceleration due to gravity.
Plugging in the values, we can calculate the shear force. Then, divide the shear force by the cross-sectional area of the bar to find the shear stress.
Step 2: Determine the vertical deflection Y of the right end of the bar (b):
The vertical deflection can be obtained using Hooke's law for linear elasticity. It is given by the formula:
Y = (F * L) / (E * A * x)
where F is the force applied, L is the length of the bar, E is the Young's modulus (also known as the elastic modulus) of the material, A is the cross-sectional area of the bar, and x is the distance from the point of application of the force to the right end of the bar.
In this case, the force applied is the weight of the crate, and x is equal to half the length of the bar.
By substituting the given values into the formula, the vertical deflection Y can be calculated.
Note: The units in the given values need to be consistent. Make sure to convert them to the appropriate units if necessary (e.g., kg to N, m2 to m).
I hope this helps! Let me know if you have any further questions.
To solve this problem, you will need to use the equation for shear stress and the equation for vertical deflection. Let's go through the steps to find the answers.
Step 1: Determine the shear stress on the bar (a):
The equation for shear stress (τ) is given by:
τ = (F/A)
where F is the force applied perpendicular to the cross-sectional area (shear force), and A is the cross-sectional area of the bar.
In this case, the force (F) is the weight of the crate, which can be calculated using the formula:
F = mg
where m is the mass of the crate and g is the acceleration due to gravity (approximately 9.8 m/s^2).
Given that the mass of the crate is 156 kg, we can calculate F:
F = (156 kg)(9.8 m/s^2) = 1528.8 N
Now, substitute the values into the shear stress equation:
τ = (1528.8 N) / (7.44 x 10^-4 m^2)
Calculate the shear stress using a calculator.
Step 2: Calculate the vertical deflection Y of the right end of the bar (b):
The equation for vertical deflection (Y) of a slender bar under the influence of an axial load is given by:
Y = (Fl) / (AE)
where F is the force applied, l is the length of the bar, A is the cross-sectional area of the bar, and E is the modulus of elasticity of the material.
In this case, the force F is the weight of the crate (1528.8 N), and l is the length of the bar (0.117 m). The cross-sectional area A is given (7.44 x 10^-4 m^2), and the modulus of elasticity E for steel is given (8.1 x 10^10 N/m^2).
Now, substitute the values into the vertical deflection equation:
Y = (1528.8 N)(0.117 m) / ((8.1 x 10^10 N/m^2)(7.44 x 10^-4 m^2))
Calculate the vertical deflection Y using a calculator.
After following these steps, you should be able to find the answers to parts (a) and (b) of the problem.