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Questions and Answers
A system consists of two particles with masses $m_1$ and $m_2$. If the velocity of the center of mass is zero, what can be said about the momenta of the two particles?
A system consists of two particles with masses $m_1$ and $m_2$. If the velocity of the center of mass is zero, what can be said about the momenta of the two particles?
- The sum of the kinetic energies of the two particles must be zero.
- The momenta of the two particles must be equal in magnitude and direction.
- The momenta of the two particles must both be zero.
- The momenta of the two particles must be equal in magnitude and opposite in direction. (correct)
A uniform metal sheet is shaped like a right-angled triangle with sides 3 cm and 4 cm. Where is the approximate location of the center of mass relative to the vertices?
A uniform metal sheet is shaped like a right-angled triangle with sides 3 cm and 4 cm. Where is the approximate location of the center of mass relative to the vertices?
- Closer to the vertex with the right angle. (correct)
- At the midpoint of the hypotenuse.
- Closer to the longest side (hypotenuse).
- At the centroid, equidistant from all vertices.
A firecracker is launched into the air and explodes into three fragments. If air resistance is negligible, what describes the motion of the center of mass of the fragments after the explosion?
A firecracker is launched into the air and explodes into three fragments. If air resistance is negligible, what describes the motion of the center of mass of the fragments after the explosion?
- It comes to an immediate stop.
- It moves in a straight line from the point of explosion.
- It follows a parabolic path as if the explosion never occurred. (correct)
- It accelerates rapidly upwards.
A system consists of two blocks of masses $m$ and $3m$ attached by a spring. Initially, the system is at rest on a frictionless surface. If the block of mass $m$ is given an initial velocity $v$, what is the maximum velocity of the block of mass $3m$?
A system consists of two blocks of masses $m$ and $3m$ attached by a spring. Initially, the system is at rest on a frictionless surface. If the block of mass $m$ is given an initial velocity $v$, what is the maximum velocity of the block of mass $3m$?
A uniform circular disc of radius R has a smaller circular hole of radius R/2 cut out of it. The center of the hole is at a distance R/2 from the center of the disc. Where is the center of mass of the resulting object?
A uniform circular disc of radius R has a smaller circular hole of radius R/2 cut out of it. The center of the hole is at a distance R/2 from the center of the disc. Where is the center of mass of the resulting object?
A person is standing on a boat that is initially at rest on a lake. The person walks from one end of the boat to the other. What happens to the boat?
A person is standing on a boat that is initially at rest on a lake. The person walks from one end of the boat to the other. What happens to the boat?
Consider a system of particles where the net external force is zero. Which of the following statements must be true?
Consider a system of particles where the net external force is zero. Which of the following statements must be true?
A rod of non-uniform density has a mass density that varies linearly from one end to the other. If the density at one end (x=0) is $\rho_0$ and at the other end (x=L) is $2\rho_0$, where is the center of mass located?
A rod of non-uniform density has a mass density that varies linearly from one end to the other. If the density at one end (x=0) is $\rho_0$ and at the other end (x=L) is $2\rho_0$, where is the center of mass located?
Two objects, one with mass m and the other with mass 2_m_, are moving with the same kinetic energy. How do their speeds compare to the center of mass speed?
Two objects, one with mass m and the other with mass 2_m_, are moving with the same kinetic energy. How do their speeds compare to the center of mass speed?
A ball is dropped from a height h onto a stationary cart. Assuming the collision is perfectly inelastic, what happens to the center of mass of the system (ball and cart) immediately after the impact?
A ball is dropped from a height h onto a stationary cart. Assuming the collision is perfectly inelastic, what happens to the center of mass of the system (ball and cart) immediately after the impact?
A square and circle of equal mass are created from a uniform metal sheet. The square has sides of length 's' and the circle has radius 'r'. Which shape has its center of mass closest to the ground when placed on an incline?
A square and circle of equal mass are created from a uniform metal sheet. The square has sides of length 's' and the circle has radius 'r'. Which shape has its center of mass closest to the ground when placed on an incline?
Two ice skaters, initially at rest, push off each other. One skater has twice the mass of the other. How do their kinetic energies compare as they move away?
Two ice skaters, initially at rest, push off each other. One skater has twice the mass of the other. How do their kinetic energies compare as they move away?
A rocket explodes in space into two unequal masses. What is true of the motion of these two masses after the explosion?
A rocket explodes in space into two unequal masses. What is true of the motion of these two masses after the explosion?
A non-uniform plank of length $L$ is found to balance at a point $L/4$ from one end. If the plank is cut into two pieces at the balance point, what is the ratio of the masses of the two pieces?
A non-uniform plank of length $L$ is found to balance at a point $L/4$ from one end. If the plank is cut into two pieces at the balance point, what is the ratio of the masses of the two pieces?
A ball is thrown vertically upwards. At the same time, a cart starts moving horizontally. Assuming the ball lands on the cart, what path will the center of mass of the ball-cart system follow?
A ball is thrown vertically upwards. At the same time, a cart starts moving horizontally. Assuming the ball lands on the cart, what path will the center of mass of the ball-cart system follow?
A system consists of three particles of equal mass located at the vertices of an equilateral triangle. Where is the center of mass of this system?
A system consists of three particles of equal mass located at the vertices of an equilateral triangle. Where is the center of mass of this system?
A wooden block is sliding down an inclined plane. If a small bird lands on the block, what will happen to the center of mass of the block-bird system?
A wooden block is sliding down an inclined plane. If a small bird lands on the block, what will happen to the center of mass of the block-bird system?
Consider a wrench is thrown in the air such that it rotates. What point on the wrench will follow a parabolic trajectory?
Consider a wrench is thrown in the air such that it rotates. What point on the wrench will follow a parabolic trajectory?
Where is the center of mass of the Earth-Moon system located?
Where is the center of mass of the Earth-Moon system located?
A car is moving at a constant velocity. A fly is buzzing around inside the car. Does the motion of the fly affect the momentum of the car?
A car is moving at a constant velocity. A fly is buzzing around inside the car. Does the motion of the fly affect the momentum of the car?
Flashcards
Center of Mass (COM)
Center of Mass (COM)
A point representing the average position of mass in a system.
COM as a Single Point
COM as a Single Point
Point where the system moves like a single particle of mass M.
R (COM Position)
R (COM Position)
The position vector of the center of mass for discrete particles.
M = Σ mᵢ
M = Σ mᵢ
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X (COM x-coordinate)
X (COM x-coordinate)
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R for Continuous Object
R for Continuous Object
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dm
dm
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Calculating COM
Calculating COM
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Continuous Objects
Continuous Objects
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V (COM Velocity)
V (COM Velocity)
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p (Total Momentum)
p (Total Momentum)
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A (COM Acceleration)
A (COM Acceleration)
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Fext
Fext
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Fext = 0
Fext = 0
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External forces
External forces
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Internal forces
Internal forces
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Mechanics Problems
Mechanics Problems
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Stability
Stability
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Astronomy
Astronomy
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Uniform Rod
Uniform Rod
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Study Notes
- The center of mass (COM) represents the average position of all the mass in a system.
- It is a theoretical point in an object or system that moves as if all mass is concentrated there.
- The system moves like a single particle of mass M at the COM.
Definition
- For n discrete particles, the center of mass is R = (Σ mᵢ*rᵢ)/M.
- R is the position vector of the center of mass.
- mᵢ is the mass of the i-th particle.
- **rᵢ is the position vector of the i-th particle.
- M = Σ mᵢ is the total mass.
- In component form:
- X = (Σ mᵢxᵢ)/M
- Y = (Σ mᵢyᵢ)/M
- Z = (Σ mᵢzᵢ)/M
- (X, Y, Z) are the coordinates of the center of mass, and (xᵢ, yᵢ, zᵢ) are the coordinates of the i-th particle.
- For a continuous object: R = (∫ r dm) / M
- r is the position vector of an infinitesimal mass element dm.
- M = ∫ dm is the total mass.
- In component form:
- X = (∫ x dm) / M
- Y = (∫ y dm) / M
- Z = (∫ z dm) / M
- For an object with uniform density ρ:
- dm = ρ dV, where dV is an infinitesimal volume element.
- R = (∫ r ρ dV) / M = (ρ ∫ r dV) / (ρ ∫ dV) = (∫ r dV) / V
- V is the total volume.
Calculating the Center of Mass
- Identify the system for which you want to find the center of mass.
- Choose a convenient coordinate system.
- For discrete particles:
- Find the mass (mᵢ) and position (rᵢ) of each particle.
- Calculate the coordinates (X, Y, Z) using summation formulas.
- For continuous objects:
- Express dm in terms of coordinates (x, y, z) and density ρ.
- Set up the integrals for X, Y, and Z.
- Evaluate the integrals to find the coordinates of the center of mass and use symmetry to simplify.
- The center of mass is at the coordinate (X, Y, Z).
Motion of the Center of Mass
- The velocity of the center of mass V:
- V = dR/dt = (Σ mᵢ drᵢ/dt) / M = (Σ mᵢvᵢ) / M
- vᵢ is the velocity of the i-th particle.
- MV = Σ mᵢ*vᵢ = p
- p is the total linear momentum of the system.
- The acceleration of the center of mass A:
- A = dV/dt = (Σ mᵢ dvᵢ/dt) / M = (Σ mᵢaᵢ) / M
- aᵢ is the acceleration of the i-th particle.
- MA = Σ mᵢ*aᵢ = Σ Fᵢ = Fₑₓₜ
- Fᵢ is the net force on the i-th particle.
- Fₑₓₜ is the net external force.
- The center of mass moves as if it were a particle of mass M subjected to the net external force.
Conservation of Momentum
- If the net external force is zero, the total linear momentum is conserved:
- Fₑₓₜ = 0 ⇒ p = constant
- MV = constant
- The velocity of the center of mass remains constant.
- This holds true even with internal forces.
- External forces are applied from outside the system (gravity, applied force).
- Internal forces are forces between objects within the system (tension, attraction).
- Newton's Third Law: internal forces cancel each other out.
Applications
- Mechanics problems: Analyzing collisions and explosions.
- Stability: Determining object stability based on the center of mass's position relative to the support base.
- Engineering: Designing stable structures and machines.
- Astronomy: Studying the motion of celestial bodies.
Examples
- Two-Particle System: For masses m₁ and m₂ at positions x₁ and x₂ on the x-axis:
- X = (m₁x₁ + m₂x₂) / (m₁ + m₂)
- Uniform Rod: For a uniform rod of length L and mass M, the center of mass is at the midpoint.
- X = L/2, assuming the rod is along the x-axis with one end at the origin.
- Triangle: The center of mass is at the intersection of its medians.
- Symmetrical Objects: For symmetrical objects with uniform density, the center of mass is at the center of symmetry.
Key Concepts
- The center of mass is a weighted average of all mass elements' positions.
- The motion is determined only by external forces.
- The concept simplifies analysis of complex systems.
- Momentum is conserved when no external forces act on the objects.
- Internal forces do not affect the overall motion.
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