Q: A uniform beam of length $$L$$ and mass $$m is $$hinged to a wall and held horizontal by a cable making an angle $$\theta$$ with the beam. If a mass $$M is $$suspended from the free end, which equation expresses the torque equilibrium about the hinge?

$$T \cdot L \cdot \sin\theta = m g \cdot \frac{L}{2} + M g \cdot L$$

Q: Which of the following is true about the second condition for equilibrium?

The sum of all torques about any axis must be zero: $$\sum \tau = 0$$

Q: A sign of mass $$M is $$suspended from the end of a uniform horizontal beam of mass $$m$$ and length $$L$$, which is attached to a wall by a hinge and supported by a cable at angle $$\theta. If $$the cable can withstand a maximum tension $$T_{max}$$, what is the largest mass $$M$$ that can be supported before the cable breaks?

$$M = \frac{T_{max} L \sin\theta - m g \frac{L}{2}}{g L}$$

Question 1

Which of the following correctly states the first condition for static equilibrium for an object?

Accepted Answer

The sum of all forces acting on the object must be zero: $$\sum F = 0$$

Question 2

A uniform beam of length $$L$$ and mass $$m is $$hinged to a wall and held horizontal by a cable making an angle $$	heta$$ with the beam. If a mass $$M is $$suspended from the free end, which equation expresses the torque equilibrium about the hinge?

Accepted Answer

$$T \cdot L \cdot \sin	heta = m g \cdot \frac{L}{2} + M g \cdot L$$

Question 3

A book rests on a table. Which pair of forces are equal in magnitude and opposite in direction, keeping the book in equilibrium?

Accepted Answer

The normal force from the table and the gravitational force on the book

Question 4

Which of the following is true about the second condition for equilibrium?

Accepted Answer

The sum of all torques about any axis must be zero: $$\sum 	au = 0$$

Question 5

A sign of mass $$M is $$suspended from the end of a uniform horizontal beam of mass $$m$$ and length $$L$$, which is attached to a wall by a hinge and supported by a cable at angle $$	heta. If $$the cable can withstand a maximum tension $$T_{max}$$, what is the largest mass $$M$$ that can be supported before the cable breaks?

Accepted Answer

$$M = \frac{T_{max} L \sin	heta - m g \frac{L}{2}}{g L}$$

Question 6

When solving a statics problem involving multiple forces and torques, which of the following is the most effective first step?

Accepted Answer

Draw a free-body diagram showing all forces and their points of application

Question 7

A horizontal uniform beam of length $$2.20\ \mathrm{m}$$ and mass $$25\ \mathrm{kg} is $$supported by a hinge at one end and a cable at $$30^\circ to $$the horizontal at the other. If a $$28\ \mathrm{kg}$$ sign hangs from the end, what is the total downward force the hinge must support?

Accepted Answer

$$(25\ \mathrm{kg} + 28\ \mathrm{kg}) \cdot 9.8\ \mathrm{m/s^2$}$$

Question 8

A mobile is suspended from a ceiling as shown: the horizontal bar is massless and extends four times as far to the right as to the left. If toy #4 (at the far right) has mass $$10\ \mathrm{g}$$, what should be the mass of toy #2 (at the far left) for equilibrium?

Accepted Answer

$$40\ \mathrm{g}$$

Question 9

Three identical balls are suspended from the ceiling by pairs of ropes of equal length, but at different angles. Which configuration will have the greatest vertical component of the tension forces in the ropes?

Accepted Answer

The configuration where the ropes are most vertical

Question 10

In a statics problem, why is the choice of axis for calculating torque arbitrary?

Accepted Answer

Because if the object is in equilibrium, the sum of torques about any axis is zero

Question 11

A ladder leans against a frictionless wall and rests on a rough floor. Which force prevents the ladder from sliding away from the wall?

Accepted Answer

The static friction force at the base of the ladder

Question 12

A uniform beam is in static equilibrium, supported at one end by a hinge and at the other by a cable. If the cable suddenly snaps, what will happen to the beam?

Accepted Answer

It will rotate downward about the hinge due to the unbalanced torque

Question 13

A sign is suspended from a horizontal beam by a cable at an angle. If the mass of the sign increases, what happens to the tension in the cable?

Accepted Answer

The tension in the cable increases

Question 14

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

Accepted Answer

Ignoring the direction of forces when writing equilibrium equations

Question 15

A beam is in static equilibrium. If the sum of the forces is zero but the sum of the torques is not zero, what will happen?

Accepted Answer

The beam will rotate

Question 16

A uniform rod of length $$L$$ and mass $$m is $$supported horizontally at its ends. If a mass $$M is $$hung at a distance $$x$$ from the left end, what is the upward force $$F_L at $$the left support?

Accepted Answer

$$F_L = \frac{(L-x)M g + \frac{L}{2} m g}{L}$$

Static Equilibrium, Elasticity, and Fracture: Study Notes

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