Part Problem I.17-19 Bending Beams If you bend a rod down, it compresses the low

Question

Part Problem I.17-19
Bending Beams If you bend a rod down, it compresses the lower side of the rod and stretches the top, resulting in a restoring force. The figure shows a beam of length L , width w, and thickness t fixed at one end and free to move at the other. Deflecting the end of the beam causes a restoring force at the end of the beam. The magnitude of the restoring force depends on the dimensions of the beam, the Young's modulus for the material, and the deflection . For small values of the deflection, the restoring force is F =[(Ywt^3)/(4L^3)]d. This is similar to the formula for the restoring force of a spring, with the quantity in brackets playing the role of the spring constant .

When a 69 man stands on the end of a springboard (a type of diving board), the board deflects by 3.0 cm.
Part A
If a 39 child stands at the end of the board, the deflection is (in cm)

0.85 .
1.7 .
2.5 .
3.4
Part B
A 69 man jumps up and lands on the end of the board, deflecting it by 12 . At this instant, what is the approximate magnitude of the upward force the board exerts on his feet? (N)

900
1800
2700
3600

Part C
If the board is replaced by one that is half the length but otherwise identical, how much will it deflect when a 69 man stands on the end?

0.38
0.75
1.5
3.0

ANSWERS WITH WORK WILL GET 5 STARS!!! I don't understand the formula..

Explanation / Answer

restoring force is F = [(Ywt^3)/(4L^3)]d.

Part a) For a given diving board, Y, w, t and L are constant. Hence, F/d is constant.

F1/d1 = F2/d2

m1*g/d1 = m2*g/d2

m1/d1 = m2/d2

Deflection with child d2 = (m2/m1)*d1 = (39/69)*3 = 1.7 cm

Part b) 69/3 = M/12

So, equivalent mass because of jumping M = 69*12/3 = 26 kg

Force = 276*9.81 = 2707 N 2700 N

Part c) F = [(Ywt^3)/(4L^3)]d.

Now, Y, w and t are constant. Hence, FL^3/d is constant.

Hence, F1*L1^3/d1 = F2*L2^3/d2

But L2 = 1/2*L1

So, F1*L1^3/d1 = F2*(L1/2)^3/d2

F1/d1 = F2/(8*d2)

Since F1 = F2 (69 kg man in both the cases), we get d2 = d1/8 = 3/8 = 0.375 cm 0.38 cm

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