The television coverage of the World Series showed several remarkable slow-motion views of the deflection of the bat as it hit the ball. Major league wood baseball bats are made of ash. A typical major league bat has a length of 0.95m and a tapering circular cross section. The ball typically hits the bat 150mm from the free end (see figure below). Occasionally, the bat breaks as it hits the ball.

For this problem, you can assume that the bat is a cylinder of constant radius and that the batter holds the end of the bat rigidly, so that it is loaded as a cantilever beam.

Derive an equation for the bending deflection δ of the bat at B, where the ball hits the bat, in terms of the applied load, P, the span, lAB, the Young's modulus of ash, E and the moment of inertia of the cross section, I.

δ =

Estimate the deflection δ of the bat at B, where it hits the ball, just before it breaks, as a function of the Young's modulus, E, and the bending strength, σb, of the ash wood, and of the span of the bat between A and B, lAB, and the bat radius, r. You can assume that the bat breaks when the maximum normal stress reaches the bending strength of ash.

Express your answer in terms of σmax,lAB,E, and r.

|δ| =

Calculate the value for the deflection δ of the bat at B, given that ash has a Young's modulus, E=10GPa and a bending strength, σb=100MPa, and that lAB=0.8m and the (assumed constant) radius of the bat is 22mm.

|δ| (in mm):

The first one:

δ = -(P*l_AB^3)/(3*E*I)

To derive the equation for the bending deflection of the bat at point B, we can apply Euler-Bernoulli beam theory. According to this theory, the deflection of a beam is related to the applied load, span, Young's modulus, and the moment of inertia of the cross-section.

The equation for the bending deflection of a cantilever beam, in this case, is given by:

δ = (P * lAB^3) / (3 * E * I)

Where:
- δ is the bending deflection of the bat at point B.
- P is the applied load (force exerted by the ball on the bat).
- lAB is the span of the bat between points A and B (distance from the free end to the point where the ball hits the bat).
- E is the Young's modulus of ash (a measure of stiffness).
- I is the moment of inertia of the bat's cross-section (related to its shape and dimensions).

Now, let's calculate the deflection |δ| of the bat at point B, just before it breaks, in terms of the Young's modulus (E), bending strength (σb), span (lAB), and bat radius (r).

The maximum normal stress (σmax) occurs at the bottom (outermost) fiber of the bat, given by:

σmax = (P * r) / I

Since the bat breaks when the maximum normal stress reaches the bending strength of ash (σb), we can equate the two:

σb = (P * r) / I

Rearranging the equation, we can solve for P:

P = (σb * I) / r

Substituting this value of P into the equation for the bending deflection, we get:

|δ| = ([(σb * I) / r] * lAB^3) / (3 * E * I)
= (σb * lAB^3) / (3 * E * r)

Now, let's calculate the value for the deflection |δ| of the bat at point B using the given parameters, where E = 10 GPa, σb = 100 MPa, lAB = 0.8 m, and the radius of the bat (r) is 22 mm.

First, we need to convert the units to be consistent:
E = 10 GPa = 10,000 MPa
r = 22 mm = 0.022 m

Substituting the values into the equation, we have:

|δ| = (σb * lAB^3) / (3 * E * r)
= (100 MPa * (0.8 m)^3) / (3 * 10,000 MPa * 0.022 m)
= 0.0933 m

Finally, we need to convert the deflection from meters to millimeters:

|δ| = 0.0933 m * 1000 mm/m
= 93.3 mm

Therefore, the deflection |δ| of the bat at point B, just before it breaks, is approximately 93.3 mm.