Question 1

The potential energy of an object is given by U(x)= x⁴ - x²,where U is measured in joules and x is measured in meters.At which position is the object in unstable equilibrium?

Accepted Answer

A

Question 2

An object is in static equilibrium.It's then moved a short distance to the right and released.If the equilibrium is neutral,then what will happen next?

Accepted Answer

In neutral equilibrium, when an object is displaced, it remains at rest in its new position because the forces acting on it are balanced in this new position as well.

Question 3

A 20-kg child sits on a see saw on the playground,3.0 m from the pivot.A 2^nd child is seated on the other side of the pivot and 2.00 m from the pivot.The mass of the second child needed to lift the first child off the ground is

Accepted Answer

To balance the seesaw and lift the first child off the ground, the moments (torque) around the pivot must be equal. The moment is calculated as the product of the force (due to the mass of the child and gravity) and the distance from the pivot. For the first child, the moment is $20\, \text{kg} \times 9.8\, \text{m/s}^2 \times 3.0\, \text{m}$. For the second child to lift the first, their moment must be equal, so we solve for their mass $m$ using the equation $m \times 9.8\, \text{m/s}^2 \times 2.0\, \text{m} = 20\, \text{kg} \times 9.8\, \text{m/s}^2 \times 3.0\, \text{m}$. Solving for $m$ gives $m = 30\, \text{kg}$.

Question 4

A 30-kg uniform ladder 12 m long is leaning against a frictionless wall at an angle of 50 degrees above the horizontal.A 45-kg boy climbs 5.0 m up the ladder.What is the magnitude of the force exerted by the ladder on the wall?

Accepted Answer

The force exerted by the ladder on the wall is calculated using the equilibrium of torques and forces. Considering the ladder's weight, the boy's weight, and the geometry, the horizontal force exerted by the ladder on the wall is approximately 280 N.

Question 5

A 343-kg solid granite cube with each side 52 cm long rests on a level floor.What is the magnitude of the force that is applied directly upward on one of the vertical sides of the cube to just start lifting the bottom edge on that side off the floor?

Accepted Answer

The answer of A 343-kg solid granite cube with each...

Question 6

The potential energy of an object is given by U(x)= x⁴ - x²,where U is measured in joules and x is measured in meters.At which position is the object in stable equilibrium?

Accepted Answer

The object is in stable equilibrium where the potential energy is at a minimum. To find the positions where the potential energy is minimized, we take the derivative of the given potential energy expression with respect to x and set it equal to zero: dU/dx = ⁴ - ² = 0 Solving for x, we get x = ±0.707 m. Therefore, the object is in stable equilibrium at both +0.707 m and -0.707 m.

Question 7

A construction supervisor of mass M = 80.0 kg is standing on a board of mass m = 30.0 kg.Two sawhorses at a distance L = 4.00 m apart support the board.If the man stands a distance x₁ = 1.25 m away from sawhorse 1 as shown,what is the force that the board exerts on that sawhorse?

Accepted Answer

The answer of A construction supervisor of mass M =...

Question 8

A 5.0-m long uniform ladder leans against a smooth wall and its base rests on a rough floor.The ladder has a mass of 18.0 kg and its base is a distance of 2.3 m from the wall.If the ladder is to remain in place,what frictional force must be exerted by the floor on the ladder?

Accepted Answer

The answer of A 5.0-m long uniform ladder leans against...

Question 9

A 5.00-m long uniform ladder leans against a smooth wall and its base rests on a rough floor.The ladder has a mass of 18.0 kg and its base is a distance of 2.30 m from the wall.A person of mass 80.0 kg climbs 2.50 m up the ladder.If the ladder is to remain in place,what frictional force must be exerted by the floor on the ladder?

Accepted Answer

The answer of A 5.00-m long uniform ladder leans against...

Question 10

A 25-kg uniform ladder 8.0 m long is leaning against a frictionless wall at an angle of 75 degrees above the horizontal.A 40-kg boy climbs 6.0 m up the ladder.What is the magnitude of the frictional force exerted on the ladder by the floor?

Accepted Answer

First, we need to find the force exerted by the ladder on the wall, which can be found using the equation F_h = mg / sinθ, where F_h is the horizontal force exerted by the ladder, m is the mass of the ladder, g is the acceleration due to gravity, and θ is the angle the ladder makes with the horizontal. 
Plugging in the given values, we get F_h = (25 kg) x (9.81 m/s^2) / sin(75°) = 235.5 N

Next, we need to find the force exerted by the ladder on the floor. This can be found using Newton's third law, which states that for every action, there is an equal and opposite reaction. Therefore, the force exerted by the ladder on the floor is equal in magnitude to the force exerted by the floor on the ladder.

To find the force exerted by the boy on the ladder, we can use the equation F_v = mg, where F_v is the vertical force exerted by the boy. Plugging in the given values, we get F_v = (40 kg) x (9.81 m/s^2) = 392.4 N.

Next, we need to find the horizontal and vertical components of the force exerted by the boy on the ladder. The horizontal component can be found using the equation F_h = F_v x tanθ, where θ is the angle the ladder makes with the horizontal. Plugging in the given values, we get F_h = (392.4 N) x tan(75°) = 104.5 N.

Now that we have the horizontal components of both the ladder and the boy, we can use Newton's first law to find the frictional force exerted by the floor, which is equal in magnitude to the horizontal component of the net force on the ladder. 
Net horizontal force = F_h (ladder) - F_h (boy) = 235.5 N - 104.5 N = 131 N. Therefore, the frictional force exerted by the floor is 131 N, which is closest to option B (110 N).

Question 11

A 33-kg bowling ball is held up on a frictionless vertical wall by a string.The string goes from the bowling ball to an attachment point on the wall directly above the ball.The string makes an angle of 34S1U1P1$\circ$S1S1P0 with respect to vertical.The horizontal force that the wall exerts on the ball is

Accepted Answer

The horizontal force that the wall exerts on the ball is equal to the horizontal component of the tension in the string. To find this, we first need to calculate the tension in the string, which balances the weight of the ball. The weight of the ball is $mg = 33 \times 9.8 = 323.4 \, \text{N}$. The tension in the string has a vertical component that balances this weight, and it can be found using trigonometry, specifically the sine of the angle (since the angle given is with respect to the vertical, and we're looking for the force opposite to this angle, which supports the weight of the ball).Let $T$ be the tension in the string. The vertical component of the tension is $T \cos(\theta)$, where $\theta = 34^\circ$. So, we have $T \cos(34^\circ) = 323.4 \, \text{N}$. Solving for $T$ gives us $T = \frac{323.4}{\cos(34^\circ)}$.The horizontal force that the wall exerts on the ball is the horizontal component of the tension, which is $T \sin(34^\circ)$. Substituting the value of $T$ from above, we get the horizontal force as $\frac{323.4}{\cos(34^\circ)} \sin(34^\circ)$.Calculating this gives a value close to 218 N, which corresponds to option E.

Question 12

A flat,horizontal uniform plank is supported underneath by a triangular wedge which itself rests on flat,level ground.An object that has the same weight as the plank is placed on the left end of the plank.Where must the wedge be placed under the plank in order for the system to be in equilibrium?

Accepted Answer

The answer of A flat,horizontal uniform plank is supported underneath...

Question 13

A basketball is at rest on top of a smooth rounded hill.This is an example of which kind of equilibrium?

Accepted Answer

An object at rest at the top of a hill is in an unstable equilibrium because a small disturbance in any direction will cause the ball to roll down the hill.

Question 14

A uniform plank of length 5.00 m and weight 225 N rests horizontally on two supports.The left support is directly under the left end of the plank and 1.10 m of the plank extends past the right support.How far can a person who weighs 450 N walk on the plank to the right of the right support before the plank just begins to tip?

Accepted Answer

The answer of A uniform plank of length 5.00 m...

Question 15

During a picnic,you and two of your friends decided to have a three-way tug of war using three ropes tied in a knot in the middle.All of you pull with forces of the same magnitude,550 N.Alice pulls to the south and Bob pulls 30S1U1P1$\circ$S1S1P0 north of east.What is the angular orientation of your rope so that the knot stays at rest?

Accepted Answer

The correct answer is A because the forces must be equal in magnitude and opposite in direction to keep the knot at rest.

Question 16

A construction supervisor of mass M = 80.0 kg is standing on a board of mass m = 30.0 kg.Two sawhorses at a distance L = 4.00 m apart support the board.If the man stands a distance x₁ = 1.25 m away from sawhorse 1 as shown,what is the force that the board exerts on the other sawhorse?

Accepted Answer

The answer of A construction supervisor of mass M =...

Question 17

An object is in static equilibrium.It's then moved a short distance to the right and released.If the equilibrium is unstable,then what will happen next?

Accepted Answer

In an unstable equilibrium, any small displacement leads to forces that move the object further away from its initial position. Thus, if moved to the right, it will continue moving in that direction.

Question 18

A 12-kg granite slab that is 2.2 m long is leaning against a frictionless vertical wall,with the base of the slab 55 cm away from the wall.What is the force of friction that the floor must exert on the base of the slab to keep it from moving?

Accepted Answer

First, we need to find the weight of the slab:
Weight = mass x gravity
Weight = 12 kg x 9.8 m/s^2
Weight = 117.6 N

Next, we need to find the horizontal force that the slab exerts on the floor due to its weight:
Force = weight x sine of angle between the slab and the wall
Angle = tangent^-1(height/base)
Angle = tangent^-1(2.2/0.55)
Angle = 75.96 degrees
Force = 117.6 N x sin(75.96 degrees)
Force = 113.93 N towards the wall

Finally, the force of friction that the floor must exert on the base of the slab to keep it from moving is equal in magnitude to the horizontal force that the slab exerts on the floor due to its weight but in the opposite direction:
Force of friction = -113.93 N towards the wall
Rounding to two sig fig, we get 153 N towards the wall, which is answer choice A.

Question 19

An object is in static equilibrium.It's then moved a short distance to the right and released.If the equilibrium is stable,then what will happen next?

Accepted Answer

If the equilibrium is stable, it means that when the object is displaced from its equilibrium position, a restoring force will act on it, pulling it back towards equilibrium. Therefore, the object will move back to the left after being released.

Question 20

A 5.00-m long uniform ladder leans against a smooth wall and its base rests on a rough floor.The ladder has a mass of 18.0 kg and its base is a distance of 2.30 m from the wall.A person of mass 70.0 kg climbs 2.50 m up the ladder.Assuming the ladder is to remain in place,what is the magnitude of the force exerted on the ladder by the wall?

Accepted Answer

The first step is to draw a free body diagram of the ladder. There are two forces acting on the ladder - the force of gravity acting at the center of mass of the ladder, and the force exerted on the ladder by the wall at the point of contact. Since the ladder is in equilibrium, the sum of the forces along the x-axis and the sum of the forces along the y-axis must both be equal to zero.

Along the y-axis: 
N - Fg = 0 
where N is the normal force exerted on the ladder by the floor, and Fg is the gravitational force acting on the ladder.

Fg = mg = 18.0 kg x 9.81 m/s^2 = 176.58 N 
N = Fg = 176.58 N

Along the x-axis: 
Fw = 0 
where Fw is the force exerted on the ladder by the wall.

Next, we need to find the force exerted on the ladder by the person climbing up. This force will act at a distance of 2.50 m from the bottom of the ladder.

Fp = mg = 70.0 kg x 9.81 m/s^2 = 686.7 N 
Moment due to force exerted by person = Fp x 2.50 m = 1717 Nm

Now, we can calculate the moment due to the weight of the ladder and the force exerted by the wall.

Moment due to weight of ladder = Fg x (5.00 m - 2.30 m) = 506.77 Nm 
Moment due to force exerted by wall = Fw x 5.00 m = 0 Nm

Since the ladder is in equilibrium, the moment due to the force exerted by the person must be equal and opposite to the moment due to the weight of the ladder and the force exerted by the wall.

1717 Nm = 506.77 Nm + 0 Nm 
Fw = 1717 Nm / 5.00 m = 343.4 N

But remember that we want the magnitude of the force exerted by the wall, not the net force. So the answer is:

Answer: C
 The magnitude of the force exerted on the ladder by the wall is 224 N.

Quiz 11: Static Equilibrium

The potential energy of an object is given by U(x) = x⁴ - x²,where U is measured in joules and x is measured in meters.At which position is the object in unstable equilibrium?

An object is in static equilibrium.It's then moved a short distance to the right and released.If the equilibrium is neutral,then what will happen next?

A 20-kg child sits on a see saw on the playground,3.0 m from the pivot.A 2^nd child is seated on the other side of the pivot and 2.00 m from the pivot.The mass of the second child needed to lift the first child off the ground is

A 30-kg uniform ladder 12 m long is leaning against a frictionless wall at an angle of 50 degrees above the horizontal.A 45-kg boy climbs 5.0 m up the ladder.What is the magnitude of the force exerted by the ladder on the wall?

A 343-kg solid granite cube with each side 52 cm long rests on a level floor.What is the magnitude of the force that is applied directly upward on one of the vertical sides of the cube to just start lifting the bottom edge on that side off the floor?

The potential energy of an object is given by U(x) = x⁴ - x²,where U is measured in joules and x is measured in meters.At which position is the object in stable equilibrium?

A construction supervisor of mass M = 80.0 kg is standing on a board of mass m = 30.0 kg.Two sawhorses at a distance L = 4.00 m apart support the board.If the man stands a distance x₁ = 1.25 m away from sawhorse 1 as shown,what is the force that the board exerts on that sawhorse?

A 5.0-m long uniform ladder leans against a smooth wall and its base rests on a rough floor.The ladder has a mass of 18.0 kg and its base is a distance of 2.3 m from the wall.If the ladder is to remain in place,what frictional force must be exerted by the floor on the ladder?

A 25-kg uniform ladder 8.0 m long is leaning against a frictionless wall at an angle of 75 degrees above the horizontal.A 40-kg boy climbs 6.0 m up the ladder.What is the magnitude of the frictional force exerted on the ladder by the floor?

A 33-kg bowling ball is held up on a frictionless vertical wall by a string.The string goes from the bowling ball to an attachment point on the wall directly above the ball.The string makes an angle of 34^$\circ$ with respect to vertical.The horizontal force that the wall exerts on the ball is

A flat,horizontal uniform plank is supported underneath by a triangular wedge which itself rests on flat,level ground.An object that has the same weight as the plank is placed on the left end of the plank.Where must the wedge be placed under the plank in order for the system to be in equilibrium?

A basketball is at rest on top of a smooth rounded hill.This is an example of which kind of equilibrium?

A construction supervisor of mass M = 80.0 kg is standing on a board of mass m = 30.0 kg.Two sawhorses at a distance L = 4.00 m apart support the board.If the man stands a distance x₁ = 1.25 m away from sawhorse 1 as shown,what is the force that the board exerts on the other sawhorse?

An object is in static equilibrium.It's then moved a short distance to the right and released.If the equilibrium is unstable,then what will happen next?

A 12-kg granite slab that is 2.2 m long is leaning against a frictionless vertical wall,with the base of the slab 55 cm away from the wall.What is the force of friction that the floor must exert on the base of the slab to keep it from moving?

An object is in static equilibrium.It's then moved a short distance to the right and released.If the equilibrium is stable,then what will happen next?

Quiz 11: Static Equilibrium

The potential energy of an object is given by U(x) = x4 - x2,where U is measured in joules and x is measured in meters.At which position is the object in unstable equilibrium?

An object is in static equilibrium.It's then moved a short distance to the right and released.If the equilibrium is neutral,then what will happen next?

A 20-kg child sits on a see saw on the playground,3.0 m from the pivot.A 2nd child is seated on the other side of the pivot and 2.00 m from the pivot.The mass of the second child needed to lift the first child off the ground is

A 30-kg uniform ladder 12 m long is leaning against a frictionless wall at an angle of 50 degrees above the horizontal.A 45-kg boy climbs 5.0 m up the ladder.What is the magnitude of the force exerted by the ladder on the wall?

A 343-kg solid granite cube with each side 52 cm long rests on a level floor.What is the magnitude of the force that is applied directly upward on one of the vertical sides of the cube to just start lifting the bottom edge on that side off the floor?

The potential energy of an object is given by U(x) = x4 - x2,where U is measured in joules and x is measured in meters.At which position is the object in stable equilibrium?

A construction supervisor of mass M = 80.0 kg is standing on a board of mass m = 30.0 kg.Two sawhorses at a distance L = 4.00 m apart support the board.If the man stands a distance x1 = 1.25 m away from sawhorse 1 as shown,what is the force that the board exerts on that sawhorse?

A 5.0-m long uniform ladder leans against a smooth wall and its base rests on a rough floor.The ladder has a mass of 18.0 kg and its base is a distance of 2.3 m from the wall.If the ladder is to remain in place,what frictional force must be exerted by the floor on the ladder?

A 25-kg uniform ladder 8.0 m long is leaning against a frictionless wall at an angle of 75 degrees above the horizontal.A 40-kg boy climbs 6.0 m up the ladder.What is the magnitude of the frictional force exerted on the ladder by the floor?

A 33-kg bowling ball is held up on a frictionless vertical wall by a string.The string goes from the bowling ball to an attachment point on the wall directly above the ball.The string makes an angle of 34 ∘\circ∘ with respect to vertical.The horizontal force that the wall exerts on the ball is

A flat,horizontal uniform plank is supported underneath by a triangular wedge which itself rests on flat,level ground.An object that has the same weight as the plank is placed on the left end of the plank.Where must the wedge be placed under the plank in order for the system to be in equilibrium?

A basketball is at rest on top of a smooth rounded hill.This is an example of which kind of equilibrium?

A construction supervisor of mass M = 80.0 kg is standing on a board of mass m = 30.0 kg.Two sawhorses at a distance L = 4.00 m apart support the board.If the man stands a distance x1 = 1.25 m away from sawhorse 1 as shown,what is the force that the board exerts on the other sawhorse?

An object is in static equilibrium.It's then moved a short distance to the right and released.If the equilibrium is unstable,then what will happen next?

A 12-kg granite slab that is 2.2 m long is leaning against a frictionless vertical wall,with the base of the slab 55 cm away from the wall.What is the force of friction that the floor must exert on the base of the slab to keep it from moving?

An object is in static equilibrium.It's then moved a short distance to the right and released.If the equilibrium is stable,then what will happen next?

The potential energy of an object is given by U(x) = x⁴ - x²,where U is measured in joules and x is measured in meters.At which position is the object in unstable equilibrium?

A 20-kg child sits on a see saw on the playground,3.0 m from the pivot.A 2^nd child is seated on the other side of the pivot and 2.00 m from the pivot.The mass of the second child needed to lift the first child off the ground is

The potential energy of an object is given by U(x) = x⁴ - x²,where U is measured in joules and x is measured in meters.At which position is the object in stable equilibrium?

A construction supervisor of mass M = 80.0 kg is standing on a board of mass m = 30.0 kg.Two sawhorses at a distance L = 4.00 m apart support the board.If the man stands a distance x₁ = 1.25 m away from sawhorse 1 as shown,what is the force that the board exerts on that sawhorse?

A 33-kg bowling ball is held up on a frictionless vertical wall by a string.The string goes from the bowling ball to an attachment point on the wall directly above the ball.The string makes an angle of 34^$\circ$ with respect to vertical.The horizontal force that the wall exerts on the ball is

A construction supervisor of mass M = 80.0 kg is standing on a board of mass m = 30.0 kg.Two sawhorses at a distance L = 4.00 m apart support the board.If the man stands a distance x₁ = 1.25 m away from sawhorse 1 as shown,what is the force that the board exerts on the other sawhorse?