Question 1

A system consists of two particles: particle 1,having the mass m₁ = 4.0 kg is located at (-4.0 m,1.0 m)and particle 2,having the mass m₂ = 5.0 kg is located at (2.0 m,1.0 m).The magnitude of the position vector of the center of mass of the system is

Accepted Answer

The position vector of the center of mass is given by:

R = (m1r1 + m2r2)/(m1 + m2)

where m1 and m2 are the masses of particles 1 and 2, and r1 and r2 are their respective position vectors.

Plugging in the given values, we get:

R = (4.0 kg)(-4.0 m,1.0 m) + (5.0 kg)(2.0 m,1.0 m)/(4.0 kg + 5.0 kg) = (-4.4 m, 1.1 m)

The magnitude of R is then:

|R| = sqrt((-4.4 m)^2 + (1.1 m)^2) = 1.2 m

Therefore, the best choice is B, with a magnitude of 1.2 m.

Question 2

A 4500-kg truck traveling due east at a speed of 101 km/h approaches a 650-kg car traveling due west at 130 km/h.The net momentum of the system consisting of the car and truck is

Accepted Answer

First, we need to convert the speeds to m/s: 
- The truck is traveling at 101 km/h = 28.06 m/s (due east) 
- The car is traveling at 130 km/h = 36.11 m/s (due west)

We can use the formula for momentum: 
p = mv

For the truck: 
p1 = 4500 kg * 28.06 m/s = 126270 kg m/s (due east)

For the car: 
p2 = 650 kg * (-36.11 m/s) = -23465.5 kg m/s (due west) (note the negative sign, since the car is traveling in the opposite direction)

The net momentum is the sum of the individual momenta: 
p = p1 + p2 
p = 126270 kg m/s - 23465.5 kg m/s 
p = 102804.5 kg m/s (due east)

Finally, we need to round to two significant figures, which gives us: 
p = 1.03 x 10^5 kg m/s (due east)

Therefore, the best choice is D.

Question 3

A toy car of mass 4.0 kg is sitting stationary when a child rolls a toy truck of mass 6.5 kg straight towards it in the negative x-direction with a speed of 5.0 m/s.The velocity of the center of mass of the system consisting of the two toys is

Accepted Answer

D

Question 4

A system consists of two particles: particle 1,having the mass m₁ = 2.0 kg has the velocity, and particle having the mass m₂ = 5.0 kg has the velocity, The magnitude of the velocity of the center of mass of the system is

Accepted Answer

The answer of A system consists of two particles: particle...

Question 5

A large number N of small identical stones are all located on the positive x-axis.The first is located at x = 1 m,the second at x = 3 m,the third at x = 5 m,and so on.The x-coordinates of all of the stones measured in meters are odd numbers.Where is their center of mass located?

Accepted Answer

The answer of A large number N of small identical...

Question 6

A system consists of two particles: particle 1,having the mass m₁ = 4.0 kg is located at (-4.0 m,1.0 m)and particle 2,having the mass m₂ = 5.0 kg is located at (2.0 m,1.0 m).The x-coordinate of the center of mass of the system is

Accepted Answer

The x-coordinate of the center of mass of a system of particles is given by:

Xcm = (m1*x1 + m2*x2 + ... + mn*xn) / (m1 + m2 + ... + mn)

where m1, m2, ..., mn are the masses of the particles and x1, x2, ..., xn are their respective x-coordinates.

Using this formula, we can find the x-coordinate of the center of mass of the given system:

Xcm = (4.0 kg * (-4.0 m) + 5.0 kg * (2.0 m)) / (4.0 kg + 5.0 kg)
    = (-16.0 kg*m + 10.0 kg*m) / 9.0 kg
    = -6.0 kg*m / 9.0 kg
    = -0.67 m

Therefore, the x-coordinate of the center of mass of the system is -0.67 m, which corresponds to choice C.

Question 7

How much force is necessary to keep a 10-cm diameter fire hose stationary if it shoots water at a velocity of 12 m/s?

Accepted Answer

The force required to keep the hose stationary is equal to the rate of change of momentum of the water. The cross-sectional area of the hose is π*(0.1/2)^2 m². The volume flow rate is area * velocity = π*(0.1/2)^2 * 12 m³/s. The mass flow rate is volume flow rate * density of water (1000 kg/m³). The force is mass flow rate * velocity = π*(0.1/2)^2 * 12 * 1000 * 12 N, which is approximately 1130 N.

Question 8

A 4350-kg truck traveling on a bridge at a velocity of (96.2 km/h,45.3 km/h)passes over a 724-kg car traveling at a velocity of (112 km/h,-35.5 km/h).The net momentum of the system consisting of the car and truck is

Accepted Answer

We can begin by finding the momentum of the truck and car separately using the equation:
momentum = mass x velocity

For the truck: 
momentum = 4350 kg x (96.2 km/h, 45.3 km/h) = (4.18 x 10^5 kg m/s, 1.97 x 10^5 kg m/s)

For the car: 
momentum = 724 kg x (112 km/h, -35.5 km/h) = (8.12 x 10^4 kg m/s, -2.57 x 10^4 kg m/s)

We can then find the net momentum of the system by adding the individual momenta together:
net momentum = (4.18 x 10^5 kg m/s, 1.97 x 10^5 kg m/s) + (8.12 x 10^4 kg m/s, -2.57 x 10^4 kg m/s) = (5.99 x 10^5 kg m/s, 1.71 x 10^5 kg m/s)

Therefore, the best choice is C, which matches the calculated net momentum.

Question 9

A thin straight wire of uniform density is oriented in the xy-plane.The ends of the wire are located at (x,y)= (-2.00 m,-1.00 m)and (x,y)= (2.00 m,3.00 m).Where is its center of mass located?

Accepted Answer

The center of mass of a uniform straight wire is located at the midpoint of the line segment connecting its ends. The midpoint can be found by averaging the x-coordinates and the y-coordinates of the ends: $x_{cm} = \frac{-2 + 2}{2} = 0$ and $y_{cm} = \frac{-1 + 3}{2} = 1$. Therefore, the center of mass is at (0, 1.00 m).

Question 10

A rocket explodes into three pieces of equal mass.Immediately after the explosion their velocities are (120 m/s,south),(240 m/s,north)and (120 m/s east).What was the velocity of the rocket's center of mass before the explosion?

Accepted Answer

First, we need to find the total momentum before the explosion, which is zero since the rocket was at rest. After the explosion, momentum is still conserved. We can use vector addition to find the velocity of the center of mass, which is the weighted average of the three pieces of the rocket based on their masses:

Total mass = 3m (since the rocket was divided into 3 equal pieces)
Velocity of the center of mass = [m(120 m/s, south) + m(240 m/s, north) + m(120 m/s, east)] / 3m
= (120/3, -120/3) + (240/3, 0) + (0, 120/3)
= (120, 0)

Therefore, the velocity of the rocket's center of mass before the explosion was 120 m/s to the right (east) and 0 m/s up/down (north/south), which gives a velocity vector of 120 m/s, northeast. The closest answer option is B.

Question 11

A toy car of mass 4.0 kg is sitting stationary when a child rolls a toy truck of mass 6.5 kg straight towards it in the negative x-direction with a speed of 5.0 m/s.What is the relative momentum?

Accepted Answer

The answer of A toy car of mass 4.0 kg...

Question 12

A toy car of mass 4.0 kg is sitting stationary when a child rolls a toy truck of mass 6.5 kg straight towards it in the negative x-direction with a speed of 5.0 m/s.The velocity of the car with respect to the center of mass of the system consisting of the two toys is

Accepted Answer

The answer of A toy car of mass 4.0 kg...

Question 13

A carbon monoxide molecule consists of a carbon atom bonded to an oxygen atom.The mass of the carbon atom is 75% the mass of the oxygen atom.The carbon atom is located at the origin and the oxygen atom is located at x = d.Where is the center of mass of the carbon monoxide molecule located?

Accepted Answer

The center of mass $x_{cm}$ of a system can be calculated using the formula $x_{cm} = \frac{\sum m_i x_i}{\sum m_i}$, where $m_i$ and $x_i$ are the mass and position of the ith particle, respectively. For the carbon monoxide molecule, let the mass of the oxygen atom be $m$, then the mass of the carbon atom is $0.75m$. The position of the carbon atom (carbon) is at the origin (0) and the position of the oxygen atom is at $d$. Substituting these values into the formula gives $x_{cm} = \frac{(0.75m \cdot 0) + (m \cdot d)}{0.75m + m} = \frac{md}{1.75m} = \frac{d}{1.75}$, which simplifies to approximately $0.57d$.

Question 14

A cylindrical bottle of oil and salt water with 1/2 volume oil ($\rho$ =0.910 g/cm³)and 1/2 volume salt water ($\rho$ = 1.03 g/cm³)is at rest on a table.Initially the oil and salt water are separated,with the oil floating on top of the water.The bottle is shaken so the oil and water uniformly mix and the bottle is returned to the table.How has the height of the center of mass changed after mixing?

Accepted Answer

The answer of A cylindrical bottle of oil and salt...

Question 15

A system consists of two particles: particle 1,having the mass m₁ = 4.0 kg is located at (-4.0 m,1.0 m)and particle 2,having the mass m₂ = 5.0 kg is located at (2.0 m,1.0 m).The direction of the position vector of the center of mass of the system measured from the positive x-axis is

Accepted Answer

The answer of A system consists of two particles: particle...

Question 16

A circular pizza of radius R has a circular piece of radius R/3 removed from one side,as shown in the figure.The center of mass of the pizza with the hole is at a distance of_____ from the center of mass of the original pizza.

Accepted Answer

The center of mass of the original pizza is at the center of the circle, since it has uniform density. 
For the pizza with the hole, we can treat it as the difference between two circles: a full circle of radius R, and a smaller circle of radius R/3. 
Using the formula for the center of mass of two overlapping circles, we can find that the center of mass of the pizza with the hole is located at a distance of (4R/3pi)/(2R/pi) = 2R/3 from the center of the circle, on the line passing through the centers of the two circles. 
The distance between the two centers of mass is therefore R - 2R/3 = R/3.

Question 17

A circular pizza of radius R has a circular piece of radius removed from one side,as shown in the figure.The center of mass of the pizza with the hole is at a distance of R/6 from the center of mass of the original pizza.The radius of the circular piece removed is

Accepted Answer

The answer of A circular pizza of radius R has...

Question 18

A cartoon character can levitate himself from the thrust obtained by firing his two revolvers down into the ground.If the bullets have a mass of 16 g and a downward velocity of 330 m/s,find his maximum weight if he can fire 12 bullets from each revolver in 10 seconds.

Accepted Answer

The maximum weight of the cartoon character is 1.3 N. This is because the thrust obtained from firing the bullets is equal to the weight of the character. The thrust is given by the equation F = mv, where F is the thrust, m is the mass of the bullets, and v is the velocity of the bullets. The weight of the character is given by the equation W = mg, where W is the weight, m is the mass of the character, and g is the acceleration due to gravity. Setting these two equations equal to each other, we get:F = Wmv = mgv = gSubstituting the given values into this equation, we get:v = 9.8 m/sThe thrust is then given by:F = mvF = (0.016 kg)(9.8 m/s)F = 0.1568 NThe weight of the character is then given by:W = FW = 0.1568 NTherefore, the maximum weight of the cartoon character is 0.1568 N, which is approximately 1.3 N.

Question 19

A long,thin rod lies along the x-axis.One end of the rod is located at x = 2.00 m and the other end of the rod is located at x = 6.00 m.The linear mass density of the rod is given by  ,where a = 0.100 kg/  and b = 0.200 kg/m.The mass of the rod is

Accepted Answer

The answer of A long,thin rod lies along the x-axis.One...

Question 20

A long,thin rod lies along the x-axis.One end of the rod is located at x = 2.00 m and the other end of the rod is located at x = 6.00 m.The linear mass density of the rod is given by  ,where a = 0.100 kg/  and b = 0.200 kg/m.What is the location of the center of mass of the rod?

Accepted Answer

The position of the center of mass of an object with variable linear mass density is given by:
x_cm = (1/M) * ∫(x * dm)
where M is the total mass of the object and dm is the linear mass density element.

In this case, M is given by:
M = ∫dm = ∫(ax + b) dx = (1/2)ax^2 + bx | 2 to 6 
M = (1/2)a(6^2 - 2^2) + b(6 - 2) 
M = 1.2 kg

To find x_cm, we need to evaluate:
x_cm = (1/M) * ∫(x * dm) 
x_cm = (1/1.2) * ∫[2 + (x - 2)(0.1)] dx | 2 to 6 
x_cm = (1/1.2) * [2(x - 2) + 0.05(x - 2)^2] | 2 to 6 
x_cm = 4.82 m

Therefore, the location of the center of mass of the rod is at x = 4.82 m, which is answer C.

Quiz 8: Systems of Particles and Extended Objects

A system consists of two particles: particle 1,having the mass m₁ = 4.0 kg is located at (-4.0 m,1.0 m) and particle 2,having the mass m₂ = 5.0 kg is located at (2.0 m,1.0 m) .The magnitude of the position vector of the center of mass of the system is

A 4500-kg truck traveling due east at a speed of 101 km/h approaches a 650-kg car traveling due west at 130 km/h.The net momentum of the system consisting of the car and truck is

A toy car of mass 4.0 kg is sitting stationary when a child rolls a toy truck of mass 6.5 kg straight towards it in the negative x-direction with a speed of 5.0 m/s.The velocity of the center of mass of the system consisting of the two toys is

A system consists of two particles: particle 1,having the mass m₁ = 2.0 kg has the velocity, and particle having the mass m₂ = 5.0 kg has the velocity, The magnitude of the velocity of the center of mass of the system is

A large number N of small identical stones are all located on the positive x-axis.The first is located at x = 1 m,the second at x = 3 m,the third at x = 5 m,and so on.The x-coordinates of all of the stones measured in meters are odd numbers.Where is their center of mass located?

A system consists of two particles: particle 1,having the mass m₁ = 4.0 kg is located at (-4.0 m,1.0 m) and particle 2,having the mass m₂ = 5.0 kg is located at (2.0 m,1.0 m) .The x-coordinate of the center of mass of the system is

How much force is necessary to keep a 10-cm diameter fire hose stationary if it shoots water at a velocity of 12 m/s?

A 4350-kg truck traveling on a bridge at a velocity of (96.2 km/h,45.3 km/h) passes over a 724-kg car traveling at a velocity of (112 km/h,-35.5 km/h) .The net momentum of the system consisting of the car and truck is

A thin straight wire of uniform density is oriented in the xy-plane.The ends of the wire are located at (x,y) = (-2.00 m,-1.00 m) and (x,y) = (2.00 m,3.00 m) .Where is its center of mass located?

A rocket explodes into three pieces of equal mass.Immediately after the explosion their velocities are (120 m/s,south) ,(240 m/s,north) and (120 m/s east) .What was the velocity of the rocket's center of mass before the explosion?

A toy car of mass 4.0 kg is sitting stationary when a child rolls a toy truck of mass 6.5 kg straight towards it in the negative x-direction with a speed of 5.0 m/s.What is the relative momentum?

A toy car of mass 4.0 kg is sitting stationary when a child rolls a toy truck of mass 6.5 kg straight towards it in the negative x-direction with a speed of 5.0 m/s.The velocity of the car with respect to the center of mass of the system consisting of the two toys is

A carbon monoxide molecule consists of a carbon atom bonded to an oxygen atom.The mass of the carbon atom is 75% the mass of the oxygen atom.The carbon atom is located at the origin and the oxygen atom is located at x = d.Where is the center of mass of the carbon monoxide molecule located?

A system consists of two particles: particle 1,having the mass m₁ = 4.0 kg is located at (-4.0 m,1.0 m) and particle 2,having the mass m₂ = 5.0 kg is located at (2.0 m,1.0 m) .The direction of the position vector of the center of mass of the system measured from the positive x-axis is

A circular pizza of radius R has a circular piece of radius R/3 removed from one side,as shown in the figure.The center of mass of the pizza with the hole is at a distance of_____ from the center of mass of the original pizza.

A circular pizza of radius R has a circular piece of radius removed from one side,as shown in the figure.The center of mass of the pizza with the hole is at a distance of R/6 from the center of mass of the original pizza.The radius of the circular piece removed is

A cartoon character can levitate himself from the thrust obtained by firing his two revolvers down into the ground.If the bullets have a mass of 16 g and a downward velocity of 330 m/s,find his maximum weight if he can fire 12 bullets from each revolver in 10 seconds.

A long,thin rod lies along the x-axis.One end of the rod is located at x = 2.00 m and the other end of the rod is located at x = 6.00 m.The linear mass density of the rod is given by ,where a = 0.100 kg/ and b = 0.200 kg/m.The mass of the rod is

A long,thin rod lies along the x-axis.One end of the rod is located at x = 2.00 m and the other end of the rod is located at x = 6.00 m.The linear mass density of the rod is given by ,where a = 0.100 kg/ and b = 0.200 kg/m.What is the location of the center of mass of the rod?

Quiz 8: Systems of Particles and Extended Objects

A system consists of two particles: particle 1,having the mass m1 = 4.0 kg is located at (-4.0 m,1.0 m) and particle 2,having the mass m2 = 5.0 kg is located at (2.0 m,1.0 m) .The magnitude of the position vector of the center of mass of the system is

A 4500-kg truck traveling due east at a speed of 101 km/h approaches a 650-kg car traveling due west at 130 km/h.The net momentum of the system consisting of the car and truck is

A toy car of mass 4.0 kg is sitting stationary when a child rolls a toy truck of mass 6.5 kg straight towards it in the negative x-direction with a speed of 5.0 m/s.The velocity of the center of mass of the system consisting of the two toys is

A system consists of two particles: particle 1,having the mass m1 = 2.0 kg has the velocity, and particle having the mass m2 = 5.0 kg has the velocity, The magnitude of the velocity of the center of mass of the system is

A large number N of small identical stones are all located on the positive x-axis.The first is located at x = 1 m,the second at x = 3 m,the third at x = 5 m,and so on.The x-coordinates of all of the stones measured in meters are odd numbers.Where is their center of mass located?

A system consists of two particles: particle 1,having the mass m1 = 4.0 kg is located at (-4.0 m,1.0 m) and particle 2,having the mass m2 = 5.0 kg is located at (2.0 m,1.0 m) .The x-coordinate of the center of mass of the system is

How much force is necessary to keep a 10-cm diameter fire hose stationary if it shoots water at a velocity of 12 m/s?

A 4350-kg truck traveling on a bridge at a velocity of (96.2 km/h,45.3 km/h) passes over a 724-kg car traveling at a velocity of (112 km/h,-35.5 km/h) .The net momentum of the system consisting of the car and truck is

A thin straight wire of uniform density is oriented in the xy-plane.The ends of the wire are located at (x,y) = (-2.00 m,-1.00 m) and (x,y) = (2.00 m,3.00 m) .Where is its center of mass located?

A rocket explodes into three pieces of equal mass.Immediately after the explosion their velocities are (120 m/s,south) ,(240 m/s,north) and (120 m/s east) .What was the velocity of the rocket's center of mass before the explosion?

A toy car of mass 4.0 kg is sitting stationary when a child rolls a toy truck of mass 6.5 kg straight towards it in the negative x-direction with a speed of 5.0 m/s.What is the relative momentum?

A toy car of mass 4.0 kg is sitting stationary when a child rolls a toy truck of mass 6.5 kg straight towards it in the negative x-direction with a speed of 5.0 m/s.The velocity of the car with respect to the center of mass of the system consisting of the two toys is

A carbon monoxide molecule consists of a carbon atom bonded to an oxygen atom.The mass of the carbon atom is 75% the mass of the oxygen atom.The carbon atom is located at the origin and the oxygen atom is located at x = d.Where is the center of mass of the carbon monoxide molecule located?

A system consists of two particles: particle 1,having the mass m1 = 4.0 kg is located at (-4.0 m,1.0 m) and particle 2,having the mass m2 = 5.0 kg is located at (2.0 m,1.0 m) .The direction of the position vector of the center of mass of the system measured from the positive x-axis is

A circular pizza of radius R has a circular piece of radius R/3 removed from one side,as shown in the figure.The center of mass of the pizza with the hole is at a distance of_____ from the center of mass of the original pizza.

A circular pizza of radius R has a circular piece of radius removed from one side,as shown in the figure.The center of mass of the pizza with the hole is at a distance of R/6 from the center of mass of the original pizza.The radius of the circular piece removed is

A cartoon character can levitate himself from the thrust obtained by firing his two revolvers down into the ground.If the bullets have a mass of 16 g and a downward velocity of 330 m/s,find his maximum weight if he can fire 12 bullets from each revolver in 10 seconds.

A long,thin rod lies along the x-axis.One end of the rod is located at x = 2.00 m and the other end of the rod is located at x = 6.00 m.The linear mass density of the rod is given by ,where a = 0.100 kg/ and b = 0.200 kg/m.The mass of the rod is

A long,thin rod lies along the x-axis.One end of the rod is located at x = 2.00 m and the other end of the rod is located at x = 6.00 m.The linear mass density of the rod is given by ,where a = 0.100 kg/ and b = 0.200 kg/m.What is the location of the center of mass of the rod?

A system consists of two particles: particle 1,having the mass m₁ = 4.0 kg is located at (-4.0 m,1.0 m) and particle 2,having the mass m₂ = 5.0 kg is located at (2.0 m,1.0 m) .The magnitude of the position vector of the center of mass of the system is

A system consists of two particles: particle 1,having the mass m₁ = 2.0 kg has the velocity, and particle having the mass m₂ = 5.0 kg has the velocity, The magnitude of the velocity of the center of mass of the system is

A system consists of two particles: particle 1,having the mass m₁ = 4.0 kg is located at (-4.0 m,1.0 m) and particle 2,having the mass m₂ = 5.0 kg is located at (2.0 m,1.0 m) .The x-coordinate of the center of mass of the system is

A system consists of two particles: particle 1,having the mass m₁ = 4.0 kg is located at (-4.0 m,1.0 m) and particle 2,having the mass m₂ = 5.0 kg is located at (2.0 m,1.0 m) .The direction of the position vector of the center of mass of the system measured from the positive x-axis is