Question 1

The linear density of a rod,in g/m,is given by $\lambda = 40.0 + 30.0 x$ .The rod extends from the origin to x = 0.400 m.What is the location of the center of mass of the rod?

Accepted Answer

The center of mass $ x_{cm} $ is given by $ x_{cm} = \frac{\int x \lambda(x) \, dx}{\int \lambda(x) \, dx} $. Integrating from 0 to 0.4 m, we find $ x_{cm} = \frac{\int_0^{0.4} x (40 + 30x) \, dx}{\int_0^{0.4} (40 + 30x) \, dx} $. Solving these integrals gives $ x_{cm} = 0.213 \, \text{m} $.

Question 2

A rocket moving in outer space maintains a constant acceleration (magnitude = 20 m/s²)while ejecting fuel at a speed of 15 km/s relative to the rocket.If the initial mass of the rocket is 3 000 kg,what is the magnitude of the thrust after 800 kg of fuel have been consumed?

Accepted Answer

The answer of A rocket moving in outer space maintains...

Question 3

When the rate of burn and the exhaust velocity are constant,a rocket ascends with

Accepted Answer

The answer of When the rate of burn and the...

Question 4

The linear density of a rod,in g/m,is given by $\lambda = 40.0 + 30.0 x$ .The rod extends from the origin to x = 0.400 m.What is the mass of the rod?

Accepted Answer

The answer of The linear density of a rod,in g/m,is...

Question 5

A rocket with an initial mass of 1 000 kg adjusts its thrust by varying the rate at which mass is ejected.The ejection speed relative to the rocket is 40 km/s.If the acceleration of the rocket is to have a magnitude of 20 m/s² at an instant when its mass is 80% of the original mass,at what rate is mass being ejected at that instant? Ignore any external forces on the rocket.

Accepted Answer

The answer of A rocket with an initial mass of...

Question 6

A 500-g firework explodes into two pieces of equal mass at an instant when it is traveling straight up at 10 m/s.If one half shoots off horizontally to the left at 20 m/s,what is the velocity,in m/s,of the other half immediately after the explosion? (The x axis is directed right;the y axis up. )

Accepted Answer

The answer of A 500-g firework explodes into two pieces...

Question 7

At an instant when a particle of mass 50 g has an acceleration of 80 m/s² in the positive x direction,a 75-g particle has an acceleration of 40 m/s² in the positive y direction.What is the magnitude of the acceleration of the center of mass of this two-particle system at this instant?

Accepted Answer

The answer of At an instant when a particle of...

Question 8

Two cars start at the same point,but travel in opposite directions on a circular path of radius R,each at speed v.While each car travels a distance less than $\frac { \pi } { 2 } \mathrm { R }$ ,one quarter circle,the center of mass of the two cars

Accepted Answer

The center of mass of the two cars moves along the diameter of the circle at a speed less than v because the cars are moving in opposite directions, reducing the net speed of the center of mass.

Question 9

Three particles are placed in the xy plane.A 40-g particle is located at (3,4)m,and a 50-g particle is positioned at (-2,-6)m.Where must a 20-g particle be placed so that the center of mass of this three-particle system is located at the origin?

Accepted Answer

The answer of Three particles are placed in the xy...

Question 10

Two boys in a canoe toss a baseball back and forth.What effect will this have on the canoe? Neglect (velocity-dependent)frictional forces with water or air.

Accepted Answer

The canoe and the boys will experience equal and opposite forces from the ball each time it is thrown. This will cause the canoe to oscillate back and forth, always moving in the opposite direction of the ball because of the conservation of momentum. However, since there are two boys in the canoe and they are both throwing the ball, the oscillation may be more complex and erratic.

Question 11

At the instant a 2.0-kg particle has a velocity of 4.0 m/s in the positive x direction,a 3.0-kg particle has a velocity of 5.0 m/s in the positive y direction.What is the speed of the center of mass of the two-particle system?

Accepted Answer

The velocity of the center of mass of a two-particle system can be found using the formula:

$v_{cm}=\frac{m_1v_1+m_2v_2}{m_1+m_2}$

Plugging in the given values, we get:

$v_{cm}=\frac{(2.0 kg)(4.0 m/s) + (3.0 kg)(5.0 m/s)}{2.0 kg + 3.0 kg}$

$v_{cm}=\frac{8.0 kg\cdot m/s + 15.0 kg\cdot m/s}{5.0 kg}$

$v_{cm}=4.6 m/s$

Therefore, the answer is B, 3.4 m/s.

Question 12

Three particles are placed in the xy plane.A 30-g particle is located at (3,4)m,and a 40-g particle is located at (-2,-2)m.Where must a 20-g particle be placed so that the center of mass of the three-particle system is at the origin?

Accepted Answer

To find the position of the third particle, we need to take the weighted average of the positions of the two particles already given, using their masses as the weights:
$$x_{cm}=\frac{m_1x_1+m_2x_2+m_3x_3}{m_1+m_2+m_3}=0$$
$$y_{cm}=\frac{m_1y_1+m_2y_2+m_3y_3}{m_1+m_2+m_3}=0$$
Substituting the given values, we get:
$$\frac{(30)(3)+(40)(-2)+20x_3}{30+40+20}=0\Rightarrow x_3=\frac{11}{2}$$
$$\frac{(30)(4)+(40)(-2)+20y_3}{30+40+20}=0\Rightarrow y_3=8$$
Therefore, the position of the third particle is (5.5,8), which corresponds to choice (E).

Question 13

Exhibit 9-1
 
Use this exhibit to answer the following question.
Two birds of prey hurtling after the same mouse collide in mid-air and grab each other with their talons.Each 250-g bird is flying at 30 m/s at a 60$\degree$ angle to the ground. 
-Refer to Exhibit 9-1.What is the horizontal component of their momentum,in $\frac { \mathrm { kg } \cdot \mathrm { m } } { \mathrm { s } }$ ,immediately after the collision?

Accepted Answer

The horizontal component of momentum is conserved in a collision. If the initial horizontal momentum was zero, it remains zero after the collision.

Question 14

Two bodies with masses m₁ and m₂ are both moving east with velocities of magnitudes v₁ and v₂,where v₁ is less than v₂.The magnitude of the velocity of the center of mass of this system of two bodies is

Accepted Answer

The velocity of the center of mass is given by Vcm=(m1v1+m2v2)/(m1+m2). Since the bodies are both moving east, the x-components of their velocities will add up and we can ignore the y-components. Thus, Vcm=(m₁v₁+m₂v₂)/(m₁+m₂). Since v₁ is less than v₂, the numerator will be less than the denominator, causing the average speed to be greater than v₁ and less than v₂. Therefore, the answer is D.

Question 15

Assume that the average mass of each of the approximately 1 billion people in China is 55 kg.Assume that they all gather in one place and climb to the top of 2 m high ladders.The center of mass of the Earth (m_E = 5.90 $x$ 10²⁴ kg)is then displaced

Accepted Answer

The answer of Assume that the average mass of each...

Question 16

At an instant when a particle of mass 80 g has a velocity of 25 m/s in the positive y direction,a 75-g particle has a velocity of 20 m/s in the positive x direction.What is the speed of the center of mass of this two-particle system at this instant?

Accepted Answer

The answer of At an instant when a particle of...

Question 17

An astronaut outside a spaceship hammers a loose rivet back in place.What happens to the astronaut as he swings the hammer?

Accepted Answer

This is due to Newton's third law of motion. When the astronaut pulls the hammer back, they exert a force on the hammer towards themselves, causing a reaction force that moves them towards the spaceship. When they swing the hammer forward, the opposite occurs: they exert a force on the hammer away from themselves, and the reaction force moves them away from the spaceship.

Question 18

A rocket engine consumes 450 kg of fuel per minute.If the exhaust speed of the ejected fuel is 5.2 km/s,what is the thrust of the rocket?

Accepted Answer

The answer of A rocket engine consumes 450 kg of...

Question 19

An exam paper contains the following equation for rocket propulsion: $( M + \Delta m ) v = M ( v + \Delta v ) + \Delta m \left( v + v _ { e } \right)$ . The error in the equation is that,instead of (v + v_e),the velocity of the fuel relative to the ground should be

Accepted Answer

The answer of An exam paper contains the following equation...

Question 20

A pitcher claims he can throw a baseball with as much momentum as a 3.00-g bullet moving with a speed of 1 500 m/s.A baseball has a mass of 0.145 kg.What must be its speed if the pitcher's claim is valid?

Accepted Answer

The answer of A pitcher claims he can throw a...

Quiz 8: Momentum and Collisions

The linear density of a rod,in g/m,is given by $\lambda = 40.0 + 30.0 x$ .The rod extends from the origin to x = 0.400 m.What is the location of the center of mass of the rod?

A rocket moving in outer space maintains a constant acceleration (magnitude = 20 m/s²) while ejecting fuel at a speed of 15 km/s relative to the rocket.If the initial mass of the rocket is 3 000 kg,what is the magnitude of the thrust after 800 kg of fuel have been consumed?

When the rate of burn and the exhaust velocity are constant,a rocket ascends with

The linear density of a rod,in g/m,is given by $\lambda = 40.0 + 30.0 x$ .The rod extends from the origin to x = 0.400 m.What is the mass of the rod?

A 500-g firework explodes into two pieces of equal mass at an instant when it is traveling straight up at 10 m/s.If one half shoots off horizontally to the left at 20 m/s,what is the velocity,in m/s,of the other half immediately after the explosion? (The x axis is directed right;the y axis up. )

At an instant when a particle of mass 50 g has an acceleration of 80 m/s² in the positive x direction,a 75-g particle has an acceleration of 40 m/s² in the positive y direction.What is the magnitude of the acceleration of the center of mass of this two-particle system at this instant?

Two cars start at the same point,but travel in opposite directions on a circular path of radius R,each at speed v.While each car travels a distance less than $\frac { \pi } { 2 } \mathrm { R }$ ,one quarter circle,the center of mass of the two cars

Three particles are placed in the xy plane.A 40-g particle is located at (3,4) m,and a 50-g particle is positioned at (-2,-6) m.Where must a 20-g particle be placed so that the center of mass of this three-particle system is located at the origin?

Two boys in a canoe toss a baseball back and forth.What effect will this have on the canoe? Neglect (velocity-dependent) frictional forces with water or air.

At the instant a 2.0-kg particle has a velocity of 4.0 m/s in the positive x direction,a 3.0-kg particle has a velocity of 5.0 m/s in the positive y direction.What is the speed of the center of mass of the two-particle system?

Three particles are placed in the xy plane.A 30-g particle is located at (3,4) m,and a 40-g particle is located at (-2,-2) m.Where must a 20-g particle be placed so that the center of mass of the three-particle system is at the origin?

Two bodies with masses m₁ and m₂ are both moving east with velocities of magnitudes v₁ and v₂,where v₁ is less than v₂.The magnitude of the velocity of the center of mass of this system of two bodies is

Assume that the average mass of each of the approximately 1 billion people in China is 55 kg.Assume that they all gather in one place and climb to the top of 2 m high ladders.The center of mass of the Earth (m_E = 5.90 $x$ 10²⁴ kg) is then displaced

At an instant when a particle of mass 80 g has a velocity of 25 m/s in the positive y direction,a 75-g particle has a velocity of 20 m/s in the positive x direction.What is the speed of the center of mass of this two-particle system at this instant?

An astronaut outside a spaceship hammers a loose rivet back in place.What happens to the astronaut as he swings the hammer?

A rocket engine consumes 450 kg of fuel per minute.If the exhaust speed of the ejected fuel is 5.2 km/s,what is the thrust of the rocket?

An exam paper contains the following equation for rocket propulsion: $( M + \Delta m ) v = M ( v + \Delta v ) + \Delta m \left( v + v _ { e } \right)$ . The error in the equation is that,instead of (v + v_e) ,the velocity of the fuel relative to the ground should be

A pitcher claims he can throw a baseball with as much momentum as a 3.00-g bullet moving with a speed of 1 500 m/s.A baseball has a mass of 0.145 kg.What must be its speed if the pitcher's claim is valid?

Quiz 8: Momentum and Collisions

The linear density of a rod,in g/m,is given by λ=40.0+30.0x\lambda = 40.0 + 30.0 xλ=40.0+30.0x .The rod extends from the origin to x = 0.400 m.What is the location of the center of mass of the rod?

A rocket moving in outer space maintains a constant acceleration (magnitude = 20 m/s2) while ejecting fuel at a speed of 15 km/s relative to the rocket.If the initial mass of the rocket is 3 000 kg,what is the magnitude of the thrust after 800 kg of fuel have been consumed?

When the rate of burn and the exhaust velocity are constant,a rocket ascends with

The linear density of a rod,in g/m,is given by λ=40.0+30.0x\lambda = 40.0 + 30.0 xλ=40.0+30.0x .The rod extends from the origin to x = 0.400 m.What is the mass of the rod?

A 500-g firework explodes into two pieces of equal mass at an instant when it is traveling straight up at 10 m/s.If one half shoots off horizontally to the left at 20 m/s,what is the velocity,in m/s,of the other half immediately after the explosion? (The x axis is directed right;the y axis up. )

At an instant when a particle of mass 50 g has an acceleration of 80 m/s2 in the positive x direction,a 75-g particle has an acceleration of 40 m/s2 in the positive y direction.What is the magnitude of the acceleration of the center of mass of this two-particle system at this instant?

Two cars start at the same point,but travel in opposite directions on a circular path of radius R,each at speed v.While each car travels a distance less than π2R\frac { \pi } { 2 } \mathrm { R }2π​R ,one quarter circle,the center of mass of the two cars

Three particles are placed in the xy plane.A 40-g particle is located at (3,4) m,and a 50-g particle is positioned at (-2,-6) m.Where must a 20-g particle be placed so that the center of mass of this three-particle system is located at the origin?

Two boys in a canoe toss a baseball back and forth.What effect will this have on the canoe? Neglect (velocity-dependent) frictional forces with water or air.

At the instant a 2.0-kg particle has a velocity of 4.0 m/s in the positive x direction,a 3.0-kg particle has a velocity of 5.0 m/s in the positive y direction.What is the speed of the center of mass of the two-particle system?

Three particles are placed in the xy plane.A 30-g particle is located at (3,4) m,and a 40-g particle is located at (-2,-2) m.Where must a 20-g particle be placed so that the center of mass of the three-particle system is at the origin?

Two bodies with masses m1 and m2 are both moving east with velocities of magnitudes v1 and v2,where v1 is less than v2.The magnitude of the velocity of the center of mass of this system of two bodies is

Assume that the average mass of each of the approximately 1 billion people in China is 55 kg.Assume that they all gather in one place and climb to the top of 2 m high ladders.The center of mass of the Earth (mE = 5.90 xxx 1024 kg) is then displaced

At an instant when a particle of mass 80 g has a velocity of 25 m/s in the positive y direction,a 75-g particle has a velocity of 20 m/s in the positive x direction.What is the speed of the center of mass of this two-particle system at this instant?

An astronaut outside a spaceship hammers a loose rivet back in place.What happens to the astronaut as he swings the hammer?

A rocket engine consumes 450 kg of fuel per minute.If the exhaust speed of the ejected fuel is 5.2 km/s,what is the thrust of the rocket?

A pitcher claims he can throw a baseball with as much momentum as a 3.00-g bullet moving with a speed of 1 500 m/s.A baseball has a mass of 0.145 kg.What must be its speed if the pitcher's claim is valid?

The linear density of a rod,in g/m,is given by $\lambda = 40.0 + 30.0 x$ .The rod extends from the origin to x = 0.400 m.What is the location of the center of mass of the rod?

A rocket moving in outer space maintains a constant acceleration (magnitude = 20 m/s²) while ejecting fuel at a speed of 15 km/s relative to the rocket.If the initial mass of the rocket is 3 000 kg,what is the magnitude of the thrust after 800 kg of fuel have been consumed?

The linear density of a rod,in g/m,is given by $\lambda = 40.0 + 30.0 x$ .The rod extends from the origin to x = 0.400 m.What is the mass of the rod?

At an instant when a particle of mass 50 g has an acceleration of 80 m/s² in the positive x direction,a 75-g particle has an acceleration of 40 m/s² in the positive y direction.What is the magnitude of the acceleration of the center of mass of this two-particle system at this instant?

Two cars start at the same point,but travel in opposite directions on a circular path of radius R,each at speed v.While each car travels a distance less than $\frac { \pi } { 2 } \mathrm { R }$ ,one quarter circle,the center of mass of the two cars

Two bodies with masses m₁ and m₂ are both moving east with velocities of magnitudes v₁ and v₂,where v₁ is less than v₂.The magnitude of the velocity of the center of mass of this system of two bodies is

Assume that the average mass of each of the approximately 1 billion people in China is 55 kg.Assume that they all gather in one place and climb to the top of 2 m high ladders.The center of mass of the Earth (m_E = 5.90 $x$ 10²⁴ kg) is then displaced