A metal meter stick made of steel rotates about its midpoint. The angular speed is slowly increased. At what value of the angular speed will the stick break apart at the center? Give your answer in rad/s.
Hint: find a relationshop between the maximum angular frequency and the breaking (ultimate tensile strength) of steal. Use the values that are given in this table in the handout of lecture 26. You also might find this example in the book to be useful here
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To find the value of the angular speed at which the metal meter stick will break apart at the center, we need to find the relationship between the maximum angular frequency and the breaking strength of steel.
The breaking strength of steel is also known as the ultimate tensile strength (UTS), which is a property that represents how much stress a material can withstand before it breaks. In this case, we need to consider the breaking strength of the metal meter stick made of steel.
To find the relationship between angular speed and breaking strength, we can use the formula for centripetal force. The centripetal force acting on the metal meter stick is equal to the tension in the stick at the point of breakage. This tension can be determined by multiplying the breaking strength of steel by the cross-sectional area of the stick.
The formula for the centripetal force is given by:
F = m * ω^2 * r
Where:
F is the centripetal force
m is the mass of the metal meter stick
ω is the angular speed
r is the radius of rotation (half of the length of the stick)
Now, we can equate the centripetal force to the tension in the stick:
F = Tension
m * ω^2 * r = Tension
To find the breaking angular speed ω at the center, we need to substitute the breaking strength of steel (UTS) for the tension:
m * ω^2 * r = UTS * A
Where:
UTS is the ultimate tensile strength of steel
A is the cross-sectional area of the stick
Now, we can rearrange the equation to solve for ω:
ω^2 = (UTS * A) / (m * r)
Taking the square root of both sides to solve for ω:
ω = sqrt((UTS * A) / (m * r))
The value of the breaking angular speed at the center (ω) will depend on the given values of UTS, A, m, and r. You can substitute the given values into the equation to calculate the specific value of ω in rad/s.