Answer: Solving Torque Problems: Shaft Diameter & Stress

[confusedkarl]

1. Homework Statement

The steel shaft of a socket wrench is 18mm diameter and 450mm long. If the allowable shear stree is 70 MN/m^2.

2. Homework Equations
i) What is the maximum permissible torque T that may be extered with the wrench?
ii) Through what angle (theta) in radians will the shaft twist under the action of the maximum torque?
iii) Through what angle (theta) in degrees will the shaft twist under the action of the maximum torque? G = 80 GN/m^2


3. The Attempt at a Solution

Im pretty happy how to do the radians to degress conversion but haven't a clue on the forumulae or process I need for i and ii =(

 

[brewnog]

For a shaft of uniform cross section:

$$\frac{T}{J} = \frac{\tau}{R} = \frac{G \theta}{l}$$

Where T is torque applied, J is the polar second moment of area, Tau is shear stress, R is radius, G is modulus of rigidity, and Theta the angle of twist. All SI units.

 

[Mene]

I would like to provide a response to the content by first addressing the importance of torque problems in engineering and mechanics. Torque is a crucial concept in understanding the motion and equilibrium of objects, especially in rotating systems. In this problem, we are given the diameter and length of a steel shaft and the allowable shear stress, and we are asked to calculate the maximum permissible torque and the corresponding angle of twist.

To solve this problem, we will use the equation T = G * theta * J/l, where T is the torque, G is the shear modulus, theta is the angle of twist, J is the polar moment of inertia, and l is the length of the shaft. We can rearrange this equation to find the angle of twist theta = T*l/(G*J). To find the maximum permissible torque, we will use the given allowable shear stress and the formula T = F*r, where F is the force applied and r is the radius of the shaft.

Using the given values, we can calculate the polar moment of inertia J = (pi*d^4)/32 = 4.99 x 10^-7 m^4, where d is the diameter of the shaft. Plugging in the values for G, J, and l, we can find the maximum permissible torque to be approximately 1.7 kNm. To find the angle of twist, we can use the formula theta = T*l/(G*J) = 0.017 radians or 0.97 degrees.

In conclusion, torque problems such as this one are essential in understanding the behavior of rotating systems and ensuring the safety and efficiency of mechanical components. By using the appropriate equations and values, we can determine the maximum permissible torque and the corresponding angle of twist for a given shaft diameter, length, and allowable shear stress.