Question 1

Equilibrium: A 20.0-kg uniform door has a width of 1.20 m and a height of 2.50 m. The door is mounted on a post by a pair of hinges, marked 1 and 2 in the figure, at the top and bottom of the door. An external force of 60.0 N, at an angle of 30.0° above the horizontal, is applied to the small doorknob, as shown in the figure. The doorknob is 1.00 m above the bottom of the door.
(a) Find the x component of the force that hinge 1 exerts on the door at the top.
(b) Find the SUM of the y components of the forces that hinges 1 and 2 together exert on the door.

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Question 2

Equilibrium: A nonuniform, 80.0-g, meterstick balances when the support is placed at the 51.0-cm mark. At what location on the meterstick should a 5.00-g tack be placed so that the stick will balance at the 50.0 cm mark?
A) 16.0 cm
B) 67.0 cm
C) 66.0 cm
D) 35.0 cm
E) 34.0 cm

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Question 3

Equilibrium: A uniform 300-kg beam, 6.00 m long, is freely pivoted at P, as shown in the figure. The beam is supported in a horizontal position by a light strut, 5.00 m long, which is freely pivoted at Q and is loosely pinned to the beam at R. A load of mass is suspended from the end of the beam at S. A maximum compression of 23,000 N in the strut is permitted, due to safety. Under maximum load, find the magnitude of the x component of the force exerted on the beam by the pivot at P. 
A) 13,800 N
B) 12,800 N
C) 11,200 N
D) 14,400 N
E) 16,000 N

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Question 4

Equilibrium: A uniform sign is supported against a wall at point P as shown in the figure. If the sign is a square  on a side and its mass is  what is the magnitude of the horizontal force that the wall at P experiences? 
A) 20 N
B) 0.00 N
C) 7.8 N
D) 98 N

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Question 5

Equilibrium: In the figure, a uniform rectangular crate 0.40 m wide and 1.0 m tall rests on a horizontal surface. The crate weighs 930 N, and its center of gravity is at its geometric center. A horizontal force F is applied at a distance h above the floor. If  what minimum value of F is required to make the crate start to tip over? Static friction is large enough that the crate does not start to slide.

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Question 6

Equilibrium: A stepladder consists of two halves, hinged at the top, and connected by a tie rod that keeps the two halves from spreading apart. In this particular instance, the two halves are 2.50 m long, the tie rod is connected to the center of each half and is 70.0 cm long. An 800-N person stands 3/5 of the way up the stepladder, as shown in the figure. Neglecting the weight of the ladder, and assuming that the ladder is resting on a smooth floor, what is the tension in the tie rod? Note: To solve this problem you must "cut" the ladder in half and consider the equilibrium of forces and torques acting on each half of the ladder. 
A) 140 N
B) 240 N
C) 280 N
D) 360 N
E) 560 N

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Question 7

Equilibrium: A 120-kg refrigerator, 2.00 m tall and 85.0 cm wide, has its center of mass at its geometrical center. You are attempting to slide it along the floor by pushing horizontally on the side of the refrigerator. The coefficient of static friction between the floor and the refrigerator is 0.300. Depending on where you push, the refrigerator may start to tip over before it starts to slide along the floor. What is the highest distance above the floor that you can push the refrigerator so that it won't tip before it begins to slide?
A) 0.710 m
B) 1.00 m
C) 1.21 m
D) 1.42 m
E) 1.63 m

Accepted Answer

For an object in static equilibrium, the sum of the forces and torques acting on the object must be equal to zero.
The weight of the refrigerator can be treated as acting at its center of mass, which is at the middle of the height of the refrigerator, or 1.00 m from the floor. The normal force from the floor will also act at this point, since the refrigerator is not accelerating vertically. 
To calculate the torque due to the weight and normal force about the point where you are pushing, we need to find the horizontal distance between that point and the line of action of each force. This distance is half the width of the refrigerator, or 0.425 m. 
The maximum static friction force that can act on the refrigerator is equal to the coefficient of static friction times the normal force, or 0.300 times the weight of the refrigerator, or 353 N. 
To avoid tipping, the torque due to your push must be equal to or less than the torque due to the weight and normal force. This means that the force you apply must be at most equal to the maximum static friction force, and the distance between your push and the line of action of the weight and normal force must be at least as great as the distance between those forces. 
We know that the line of action of the weight and normal force is at 1.00 m from the floor, and half the width of the refrigerator away from the point where you push. So this distance is 0.425 m. 
If you push at a height of h above the floor, the torque due to your push is equal to your force times the horizontal distance between your push and the center of mass. This distance is equal to the height h minus half the height of the refrigerator, or (h - 1.00m). 
So we have the inequality: 
F_push × (h - 1.00m) ≤ 353 N × 0.425 m 
where F_push is the force you apply. 
We also know that the force you apply must be less than or equal to the maximum static friction force: 
F_push ≤ 353 N 
Combining these inequalities gives: 
353 N × (h - 1.00m) ≤ 353 N × 0.425 m 
h ≤ 1.00m + (0.425 m/353 N) 
h ≤ 1.427 m 
Therefore, the highest distance above the floor that you can push the refrigerator so that it won't tip before it begins to slide is 1.42 m.

Question 8

Equilibrium: A 5.0-m long, 12-kg uniform ladder rests against a smooth vertical wall with the bottom of the ladder 3.0 m from the wall. The coefficient of static friction between the floor and the ladder is 0.28. What distance, measured along the ladder from the bottom, can a 60-kg person climb before the ladder starts to slip?
A) 4.0 m
B) 3.7 m
C) 1.7 m
D) 1.3 m
E) 3.3 m

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Question 9

Equilibrium: A 30.0-kg child sits on one end of a long uniform beam having a mass of 20.0 kg, and a 40.0-kg child sits on the other end. The beam balances when a fulcrum is placed below the beam a distance 1.10 m from the 30.0-kg child. How long is the beam?
A) 2.12 m
B) 1.98 m
C) 1.93 m
D) 2.07 m
E) 2.20 m

Accepted Answer

To find the length of the beam, we can use torque equilibrium. Since the beam is in equilibrium, we know that the sum of the clockwise torques around the fulcrum must equal the sum of the counterclockwise torques. We can choose the fulcrum as the point around which we calculate the torques. Let's take the torque due to the 30.0 kg child as positive (counterclockwise) and the torque due to the 40.0 kg child as negative (clockwise). Then we have:

$(30.0	ext{ kg})(g)(1.10	ext{ m}) - (40.0	ext{ kg})(g)(L - 1.10	ext{ m}) = 0$

where $g$ is the acceleration due to gravity and $L$ is the length of the beam. Solving for $L$, we get:

$L = \frac{(30.0	ext{ kg})(g)(1.10	ext{ m})}{(40.0	ext{ kg})(g)} + 1.10	ext{ m} = 1.98	ext{ m}$

Therefore, the best choice is (B) 1.98 m.

Question 10

Equilibrium: A solid uniform brick is placed on a sheet of wood. When one end of the sheet is raised (see figure), you observe that the maximum that the angle θ can be without tipping over the brick is 49.6°. There is enough friction to prevent the brick from sliding. What is the width w of the brick? 
A) 5.18 cm
B) 6.09 cm
C) 6.81 cm
D) 9.40 cm
E) 10.5 cm

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Question 11

Equilibrium: In the figure, a 10.0-m long bar is attached by a frictionless hinge to a wall and held horizontal by a rope that makes an angle θ of 53° with the bar. The bar is uniform and weighs 39.9 N. How far from the hinge should a 10.0-kg mass be suspended for the tension T in the rope to be 125 N?

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Question 12

Equilibrium: In the figure, the horizontal lower arm has a mass of 2.8 kg and its center of gravity is 12 cm from the elbow joint pivot. How much force F_M must the vertical extensor muscle in the upper arm exert on the lower arm to hold a 7.5 kg shot put? A) 100 N B) 500 N C) 750 N D) 1000 N E) 1500 N

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Question 13

Equilibrium: In the figure, a uniform ladder 12 meters long rests against a vertical frictionless wall. The ladder weighs 400 N and makes an angle θ of 79° with the floor. A man weighing 790 N climbs slowly up the ladder When he has climbed to a point that is 7.8 m from the base of the ladder, the ladder starts to slip. What is the coefficient of static friction between the floor and the ladder?

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Question 14

Equilibrium: A child is trying to stack two uniform wooden blocks, 12 cm in length, so they will protrude as much as possible over the edge of a table, without tipping over, as shown in the figure. What is the maximum possible overhang distance d? 
A) 5 cm
B) 6 cm
C) 7 cm
D) 8 cm
E) 9 cm

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Question 15

Equilibrium: A 3.00-kg ball rests in a frictionless groove as shown in the figure.  (a) What is the magnitude of the force that the left side of the groove exerts on the ball?
(b) What is the magnitude of the force that the right side of the groove exerts on the ball?

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Question 16

Equilibrium: A uniform 300-kg beam, 6.00 m long, is freely pivoted at P, as shown in the figure. The beam is supported in a horizontal position by a light strut, 5.00 m long, which is freely pivoted at Q and is loosely pinned to the beam at R. A load of mass is suspended from the end of the beam at S. A maximum compression of 23,000 N in the strut is permitted, due to safety. The maximum mass M of the load is closest to 
A) 789 kg.
B) 554 kg.
C) 1020 kg.
D) 1090 kg.
E) 1320 kg.

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Question 17

Equilibrium: A 10.0-kg uniform ladder that is 2.50 m long is placed against a smooth vertical wall and reaches to a height of 2.10 m, as shown in the figure. The base of the ladder rests on a rough horizontal floor whose coefficient of static friction with the ladder is 0.800. An 80.0-kg bucket of concrete is suspended from the top rung of the ladder, right next to the wall, as shown in the figure. What is the magnitude of the friction force that the floor exerts on the ladder? 
A) 538 N
B) 706 N
C) 1290 N
D) 833 N
E) 601 N

Accepted Answer

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Question 18

Equilibrium: A dump truck has a large cubical concrete block in its bed. The coefficients of friction between this block and the floor of the bed are µ_k = 0.450 and µ_s = 0.650. As the bed is slowly tilted above the horizontal, will the brick first begin to slide or will it first tip over? A) It will first tip over. B) It will first begin to slide. C) It will tip over just as it begins to slide. D) It is impossible to answer without knowing the mass of the block. E) It is impossible to answer without knowing the dimensions of the block.

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Question 19

Equilibrium: Two identical ladders are 3.0 m long and weigh 600 N each. They are connected by a hinge at the top and are held together by a horizontal rope, 1.0 m above the smooth floor forming a symmetric "A" arrangement. The angle between the ladders is 60° and both ladders have their center of gravity at their midpoint. What is the tension in the rope?
A) 240 N
B) 300 N
C) 220 N
D) 260 N
E) 280 N

Accepted Answer

The tension in the rope can be found by analyzing the forces acting on one of the ladders and using the equilibrium conditions. The center of gravity of each ladder acts at its midpoint, which is 1.5 m from the top (since the ladders are 3.0 m long). The horizontal component of the force due to gravity (which is what the rope must counteract to keep the ladder in equilibrium) can be found using trigonometry. The force due to gravity on one ladder is 600 N, and this force acts downward at the ladder's midpoint. The angle between the ladder and the floor is 30° (since the ladders form a 60° angle at the top and are symmetric, each ladder makes a 30° angle with the vertical). The horizontal component of the gravitational force is $600 N \times \sin(30°) = 600 N \times 0.5 = 300 N$. However, this force is shared equally by the two ropes (one on each side of the "A" arrangement), so the tension in one rope is $300 N / 2 = 150 N$. This calculation error led to the incorrect conclusion. The correct approach involves calculating the tension based on the equilibrium of torques around the point where a ladder touches the floor.To correctly solve for the tension, consider the torque about the base of the ladder due to the weight of the ladder and the tension in the rope. The perpendicular distance from the line of action of the weight to the pivot (base of the ladder) is $1.5 m \times \cos(30°)$. The force due to gravity is 600 N acting at the midpoint. The tension T in the rope creates a torque that counteracts the torque created by the weight of the ladder. The perpendicular distance from the rope to the pivot is 1.0 m (given). Setting the torques equal for equilibrium (and solving for T) involves using the relationship between the components of forces and the distances they act over. The correct calculation involves analyzing the system more thoroughly, considering the forces and torques in equilibrium, and applying trigonometry correctly. My initial explanation mistakenly simplified the problem and did not account for the correct application of equilibrium conditions and torque. The correct tension should be calculated by considering the entire system's equilibrium, including both the forces and the torques acting on the ladders.

Quiz 8: Momentum, Impulse, and Collisions

Equilibrium: A nonuniform, 80.0-g, meterstick balances when the support is placed at the 51.0-cm mark. At what location on the meterstick should a 5.00-g tack be placed so that the stick will balance at the 50.0 cm mark?

Equilibrium: A uniform sign is supported against a wall at point P as shown in the figure. If the sign is a square on a side and its mass is what is the magnitude of the horizontal force that the wall at P experiences?

Equilibrium: A 30.0-kg child sits on one end of a long uniform beam having a mass of 20.0 kg, and a 40.0-kg child sits on the other end. The beam balances when a fulcrum is placed below the beam a distance 1.10 m from the 30.0-kg child. How long is the beam?

Equilibrium: In the figure, the horizontal lower arm has a mass of 2.8 kg and its center of gravity is 12 cm from the elbow joint pivot. How much force FM must the vertical extensor muscle in the upper arm exert on the lower arm to hold a 7.5 kg shot put?

Equilibrium: A child is trying to stack two uniform wooden blocks, 12 cm in length, so they will protrude as much as possible over the edge of a table, without tipping over, as shown in the figure. What is the maximum possible overhang distance d?

Equilibrium: A 3.00-kg ball rests in a frictionless groove as shown in the figure. (a) What is the magnitude of the force that the left side of the groove exerts on the ball? (b) What is the magnitude of the force that the right side of the groove exerts on the ball?

Equilibrium: A dump truck has a large cubical concrete block in its bed. The coefficients of friction between this block and the floor of the bed are µk = 0.450 and µs = 0.650. As the bed is slowly tilted above the horizontal, will the brick first begin to slide or will it first tip over?

Equilibrium: In the figure, the horizontal lower arm has a mass of 2.8 kg and its center of gravity is 12 cm from the elbow joint pivot. How much force F_M must the vertical extensor muscle in the upper arm exert on the lower arm to hold a 7.5 kg shot put?

Equilibrium: A dump truck has a large cubical concrete block in its bed. The coefficients of friction between this block and the floor of the bed are µ_k = 0.450 and µ_s = 0.650. As the bed is slowly tilted above the horizontal, will the brick first begin to slide or will it first tip over?