Centre of Mass Quiz

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Questions and Answers

When two particles have the same mass, where does the center of mass lie?

Closer to the lighter particle

Exactly midway between the two particles (correct)

At the origin of the coordinate system

At the position of the heavier particle

What is the formula for the total mass (M) of a system of n particles?

M = ∑ mi / n

M = n * mi

M = ∑ mi (correct)

M = ∑ (mi * xi)

How is the position of the center of mass (X) defined for a system of n particles?

X = ∑ (xi) / n

X = ∑ (mi * yi) / M

X = ∑ (xi * M) / n

X = ∑ (mi * xi) / ∑ mi (correct)

In a system of multiple disparate mass particles, which statement about their center of mass is true?

It considers the positions of all particles relative to their masses Signup and view all the answers

What happens to the position of the center of mass if one particle is significantly heavier than the others?

It shifts closer to the heavier particle Signup and view all the answers

If masses m1, m2, ..., mn are distributed along the x-axis, what is required to calculate the x-coordinate of the center of mass?

The individual x-coordinates and masses of all particles Signup and view all the answers

What is the force acting on the first particle in a system of particles?

F1 = m1a1 Signup and view all the answers

Which equation represents the position X of the center of mass for n particles?

X = ∑ (mi * xi) / M Signup and view all the answers

How is the acceleration of the centre of mass represented?

A = d V / dt Signup and view all the answers

What does the term xi represent in the context of calculating the center of mass?

The x-coordinate of the ith particle Signup and view all the answers

What does Newton's second law state in the context of a system of particles?

The acceleration of the centre of mass is determined by the external forces acting at that point. Signup and view all the answers

What assumption was made in earlier studies regarding internal motion of the particles?

Internal motion was negligible or absent. Signup and view all the answers

In the equation MA = F1 + F2 + ... + Fn, what does MA represent?

Mass of the whole system multiplied by acceleration. Signup and view all the answers

What is the representation of the linear momentum of the first particle?

m1v1 Signup and view all the answers

Why was the procedure for analyzing forces on bodies previously not explicitly outlined?

It was deemed unnecessary given the simplicity of cases. Signup and view all the answers

What is not needed in the current treatment of particles as systems?

Assumption of negligible internal interactions. Signup and view all the answers

What do Px, Py, and Pz represent in the context of momentum?

The components of total linear momentum Signup and view all the answers

In the absence of external forces, how does the center of mass of a double star system behave?

It moves like a free particle Signup and view all the answers

Why is it often convenient to work in the center of mass frame in particle systems?

It often reveals easier trajectories of particles Signup and view all the answers

In the context of binary stars, how do their trajectories appear when viewed from the center of mass frame?

They look complicated Signup and view all the answers

When a heavy nucleus splits into lighter nuclei, what happens to the center of mass?

It can either be in uniform motion or at rest Signup and view all the answers

What is the typical relationship between two stars in a binary system regarding their center of mass?

They orbit around a common center of mass Signup and view all the answers

What is indicated by the statement that the two product particles in a splitting nucleus fly back to back?

Their momenta are equal and opposite Signup and view all the answers

How do the trajectories of stars in a binary system appear in a laboratory frame of reference?

They exhibit complex motion patterns Signup and view all the answers

What characterizes a rigid body in the context of particle systems?

It behaves as if it's a continuous mass distribution. Signup and view all the answers

In calculating the center of mass for a rigid body with equal masses, which equation applies?

$Y = \frac{m_1 y_1 + m_2 y_2 + m_3 y_3}{m_1 + m_2 + m_3}$ Signup and view all the answers

How can the center of mass be approximated for a continuous distribution of a rigid body?

By using the positions of discrete mass elements. Signup and view all the answers

When considering a thin rod, which characteristic can be assumed about its mass distribution?

Its width can be neglected in comparison to its length. Signup and view all the answers

What symmetry principle is applied when determining the positions of particles in a thin rod?

Opposite elements on the rod equally balance each other out. Signup and view all the answers

What happens when the distance between particles in a rigid body is considered?

The calculations become simpler for determining the center of mass. Signup and view all the answers

What is indicated about the particles in a rigid body when using equations for center of mass?

They can be approximated as a collection rather than individually. Signup and view all the answers

What is essential when subdividing a rigid body into small mass elements for calculating its center of mass?

The mass of elements must be balanced across the body. Signup and view all the answers

What happens to the total linear momentum of a system during the radioactive decay of a moving unstable particle?

It remains the same. Signup and view all the answers

In the radioactive decay of a radium nucleus, into which particles does it disintegrate?

Radon nucleus and an alpha particle. Signup and view all the answers

From what frame of reference does the motion of decay particles look particularly simple?

The frame where the centre of mass is at rest. Signup and view all the answers

Which of the following describes the forces that lead to the decay of a nucleus?

Forces leading to decay are internal to the system. Signup and view all the answers

When analyzing the motion of particles after decay, what path does the center of mass follow?

It moves along the path of the original decaying nucleus. Signup and view all the answers

What is the significance of separating the motion of different parts of a system into two components?

It simplifies understanding of the system's motion. Signup and view all the answers

Which vector product is defined as a scalar product of two vector quantities?

Dot product. Signup and view all the answers

How are the trajectories of the stars described in relation to the centre of mass of a system?

They are a combination of uniform motion and circular orbits. Signup and view all the answers

Study Notes

Centre of Mass

The centre of mass of a system of particles is a point where the total mass of the system can be considered to be concentrated.
For two particles of equal mass (m1 = m2 = m), the centre of mass lies exactly midway between them.
For n particles with different masses (m1, m2, … mn) along a straight line, the position of the centre of mass is given by the following formula:
- X = (m1x1 + m2x2 + … + mnxn) / (m1 + m2 + … + mn)
For a system of n particles in space, the coordinates of the centre of mass are given by:
- X = (∑ mi xi) / M
- Y = (∑ mi yi) /M
- Z = (∑ mi zi) / M
The total mass of the system is given by M = ∑ mi.
The position of the ith particle is given by (xi, yi, zi).
These equations are applicable to a rigid body, which can be treated as a continuous distribution of mass when the number of particles is very large.

Motion of the Centre of Mass

Newton’s second law of motion states that the force acting on a particle is equal to its mass times its acceleration. (F = ma)
The force acting on a system of particles can be calculated by summing up the forces acting on each individual particle.
The acceleration of the centre of mass (A) of a system of particles is given by:
- A = (∑ mi ai) / M
The total force on the system (F) is the sum of the forces acting on individual particles:
- F = F1 + F2 + … + Fn
- Where F1, F2, … Fn are the forces acting on the first, second, … nth particle, respectively.
Therefore, applying Newton’s second law to the entire system:
- MA = F1 + F2 + … + Fn

Linear Momentum

The linear momentum of a particle is its mass times its velocity (p = mv).
The total linear momentum of a system of particles is the vector sum of the linear momenta of the individual particles.
The total linear momentum of a system of particles is given by:
- P = m1v1 + m2v2 + … + mnv n
- Where v1, v2, … vn are the velocities of the first, second, … nth particle, respectively.
Applying Newton’s second law to the total linear momentum, we get:
- dP/dt = F
This equation expresses the conservation of linear momentum: the total linear momentum of a system remains constant if the net external force acting on the system is zero.

Conservation of Linear Momentum

The total linear momentum of an isolated system remains constant.
In a closed system, where no external forces act, the total linear momentum before and after an interaction remains constant.
Examples of conservation of linear momentum include:
- The motion of a rocket after it expels fuel.
- The collision of two particles.
- Radioactive decay of a nucleus.

Centre of Mass Frame of Reference

It is often convenient to work in the centre of mass frame of reference when analysing the motion of a system of particles.
In this frame, the centre of mass of the system is at rest.
The motion of the particles in the centre of mass frame can be used to understand the motion of the system as a whole.
Examples include:
- The motion of binary stars.
- The decay of a radioactive nucleus.

Vector Product of Two Vectors

The vector product of two vectors is a vector quantity that is perpendicular to both of the original vectors.
It is denoted by the symbol x and is calculated using the following formula:
- a x b = |a||b|sinθ n
- Where a and b are two vectors, θ is the angle between them, and n is a unit vector perpendicular to both a and b.
The direction of the vector product is determined using the right-hand rule.

Right Hand Rule

To apply the right-hand rule:
- Point the fingers of your right hand in the direction of the first vector.
- Curl your fingers towards the direction of the second vector.
- Your thumb will now be pointing in the direction of the vector product.

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Test your understanding of the centre of mass and its calculations with this quiz. Discover how the position of the centre of mass varies with different particle configurations and learn the applicable formulas. Ideal for students diving into physics concepts.

Centre of Mass Quiz

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Questions and Answers

When two particles have the same mass, where does the center of mass lie?

What is the formula for the total mass (M) of a system of n particles?

How is the position of the center of mass (X) defined for a system of n particles?

In a system of multiple disparate mass particles, which statement about their center of mass is true?

What happens to the position of the center of mass if one particle is significantly heavier than the others?

If masses m1, m2, ..., mn are distributed along the x-axis, what is required to calculate the x-coordinate of the center of mass?

What is the force acting on the first particle in a system of particles?

Which equation represents the position X of the center of mass for n particles?

How is the acceleration of the centre of mass represented?

What does the term xi represent in the context of calculating the center of mass?

What does Newton's second law state in the context of a system of particles?

What assumption was made in earlier studies regarding internal motion of the particles?

In the equation MA = F1 + F2 + ... + Fn, what does MA represent?

What is the representation of the linear momentum of the first particle?

Why was the procedure for analyzing forces on bodies previously not explicitly outlined?

What is not needed in the current treatment of particles as systems?

What do Px, Py, and Pz represent in the context of momentum?

In the absence of external forces, how does the center of mass of a double star system behave?

Why is it often convenient to work in the center of mass frame in particle systems?

In the context of binary stars, how do their trajectories appear when viewed from the center of mass frame?

When a heavy nucleus splits into lighter nuclei, what happens to the center of mass?

What is the typical relationship between two stars in a binary system regarding their center of mass?

What is indicated by the statement that the two product particles in a splitting nucleus fly back to back?

How do the trajectories of stars in a binary system appear in a laboratory frame of reference?

What characterizes a rigid body in the context of particle systems?

In calculating the center of mass for a rigid body with equal masses, which equation applies?

How can the center of mass be approximated for a continuous distribution of a rigid body?

When considering a thin rod, which characteristic can be assumed about its mass distribution?

What symmetry principle is applied when determining the positions of particles in a thin rod?

What happens when the distance between particles in a rigid body is considered?

What is indicated about the particles in a rigid body when using equations for center of mass?

What is essential when subdividing a rigid body into small mass elements for calculating its center of mass?

What happens to the total linear momentum of a system during the radioactive decay of a moving unstable particle?

In the radioactive decay of a radium nucleus, into which particles does it disintegrate?

From what frame of reference does the motion of decay particles look particularly simple?

Which of the following describes the forces that lead to the decay of a nucleus?

When analyzing the motion of particles after decay, what path does the center of mass follow?

What is the significance of separating the motion of different parts of a system into two components?

Which vector product is defined as a scalar product of two vector quantities?

How are the trajectories of the stars described in relation to the centre of mass of a system?

Study Notes

Centre of Mass

Motion of the Centre of Mass

Linear Momentum

Conservation of Linear Momentum

Centre of Mass Frame of Reference

Vector Product of Two Vectors

Right Hand Rule

Studying That Suits You

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