Q: A uniform beam of length $$L$$ and mass $$m is $$supported horizontally by a hinge at one end and a cable at the other end, which makes an angle $$\theta$$ with the beam. If a mass $$M is $$suspended from the end of the beam, what is the tension $$T in $$the cable when the system is in equilibrium?

$$T = \dfrac{(M + \frac{1}{2}m)g}{\sin\theta}$$

Q: A 5.0-m-long ladder of mass 12.0 kg leans against a frictionless wall at a height of 4.0 m above the ground. What is the magnitude of the normal force exerted by the ground on the ladder?

$$F_N = 12.0 \times 9.8 = 117.6\ \mathrm{N}$$

Q: Which of the following best describes the second condition for equilibrium?

The net torque about any axis must be zero: $$\sum \tau = 0$$

Q: A steel wire of length $$1.60\ \mathrm{m}$$ and diameter $$0.20\ \mathrm{cm}$$ stretches by $$0.25\ \mathrm{cm}$$ under tension. If Young's modulus for steel is $$2.0 \times 10$$^{11}$$\ \mathrm{N/m^2$}$$, what is the tension in the wire?

$$T = 1.57 \times 10^3$\ \mathrm{N}$$

Q: A bar of length $$0.5\ \mathrm{m} is $$subjected to a stress of $$1.0 \times 10^6$\ \mathrm{N/m^2$}. If $$Young's modulus is $$2.0 \times 10$$^{11}$$\ \mathrm{N/m^2$}$$, what is the change in length?

$$2.5 \times 10$$^{-6}$$\ \mathrm{m}$$

Question 1

A uniform beam of length $$L$$ and mass $$m is $$supported horizontally by a hinge at one end and a cable at the other end, which makes an angle $$	heta$$ with the beam. If a mass $$M is $$suspended from the end of the beam, what is the tension $$T in $$the cable when the system is in equilibrium?

Accepted Answer

$$T = \dfrac{(M + \frac{1}{2}m)g}{\sin	heta}$$

Question 2

A 5.0-m-long ladder of mass 12.0 kg leans against a frictionless wall at a height of 4.0 m above the ground. What is the magnitude of the normal force exerted by the ground on the ladder?

Accepted Answer

$$F_N = 12.0 	imes 9.8 = 117.6\ \mathrm{N}$$

Question 3

Which of the following best describes the second condition for equilibrium?

Accepted Answer

The net torque about any axis must be zero: $$\sum 	au = 0$$

Question 4

A steel wire of length $$1.60\ \mathrm{m}$$ and diameter $$0.20\ \mathrm{cm}$$ stretches by $$0.25\ \mathrm{cm}$$ under tension. If Young's modulus for steel is $$2.0 	imes 10$$^{11}$$\ \mathrm{N/m^2$}$$, what is the tension in the wire?

Accepted Answer

$$T = 1.57 	imes 10^3$\ \mathrm{N}$$

Question 5

A bar of length $$L$$ and cross-sectional area $$A is $$subjected to a tensile force $$F$$, causing an elongation $$\Delta L. $$Which equation correctly relates Young's modulus $$E to $$these quantities?

Accepted Answer

$$E = \dfrac{F L}{A \Delta L}$$

Question 6

A gymnast balances on a beam. If she shifts her center of gravity outside the base of support, what type of equilibrium does she experience?

Accepted Answer

Unstable equilibrium

Question 7

A 3.0 m long steel rod experiences a strain of $$0.001$$ under tension. By how much does its length change?

Accepted Answer

$$0.003\ \mathrm{m}$$

Question 8

Which of the following statements about the center of gravity is correct?

Accepted Answer

For a symmetrical object, the center of gravity lies at its geometric center.

Question 9

A cable is $$100\ \mathrm{m}$$ long and has a cross-sectional area of $$1.0\ \mathrm{mm}^2. A$$ $$1000\ \mathrm{N}$$ force is applied to stretch the cable. The elastic modulus for the cable is $$1.0 	imes 10$$^{11}$$\ \mathrm{N/m^2$}. $$How far does it stretch?

Accepted Answer

$$0.001\ \mathrm{m}$$

Question 10

Which of the following best describes the elastic limit of a material?

Accepted Answer

The maximum stress that can be applied before permanent deformation occurs.

Question 11

A bone is subjected to a compressive force. Which region of the stress-strain curve represents permanent deformation?

Accepted Answer

Plastic region

Question 12

A uniform rod of mass $$m$$ and length $$L is $$attached to a wall and held horizontally. What is the torque due to gravity about the point of attachment?

Accepted Answer

$$	au = -mg\dfrac{L}{2}$$

Question 13

Which of the following is NOT a correct step in solving statics problems?

Accepted Answer

Assume the net force is nonzero.

Question 14

A bar of length $$0.5\ \mathrm{m} is $$subjected to a stress of $$1.0 	imes 10^6$\ \mathrm{N/m^2$}. If $$Young's modulus is $$2.0 	imes 10$$^{11}$$\ \mathrm{N/m^2$}$$, what is the change in length?

Accepted Answer

$$2.5 	imes 10$$^{-6}$$\ \mathrm{m}$$

Question 15

Which of the following is the correct definition of normal stress?

Accepted Answer

$$\sigma = \dfrac{F}{A}$$

Question 16

A person stands on one leg. The force exerted by the biceps muscle to hold a weight in the hand is much greater than the weight itself. Why?

Accepted Answer

Because the muscle attaches close to the joint, requiring a larger force to balance the torque.

Question 17

Which of the following statements about stable equilibrium is correct?

Accepted Answer

If displaced, the object tends to return to its original position.

Question 18

A material is loaded beyond its proportional limit but below its breaking point. What happens?

Accepted Answer

It undergoes permanent deformation.

Question 19

Which of the following is the correct expression for the spring constant $$k of a $$bar in terms of Young's modulus $$E$$, cross-sectional area $$A$$, and length $$L_0$$?

Accepted Answer

$$k = \dfrac{EA}{L_0$}$$

Question 20

A rod is in static equilibrium. Which of the following must be true?

Accepted Answer

$$\sum F_x = 0$$, $$\sum F_y = 0$$, and $$\sum 	au = 0$$

Chapter 12: Static Equilibrium, Elasticity, and Fracture – Study Notes

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