Podcast Beta
Questions and Answers
What is the formula to calculate the center of mass for two bodies?
In a perfectly inelastic collision, what can be said about the final momentum?
What does the variable 'e' represent in the context of collision?
Which equation represents the condition for no movement of the center of mass in a two-body system?
Signup and view all the answers
How is the acceleration of the center of mass represented in a pulley block system?
Signup and view all the answers
What is the principle of conservation of momentum for a closed system during an explosion?
Signup and view all the answers
Which expression represents the distance of the center of mass for a continuous mass system?
Signup and view all the answers
What does the variable ΔKE represent in an inelastic collision?
Signup and view all the answers
Study Notes
Center of Mass
- The center of mass (Com) is the average position of all the mass points in a system.
- To find the Com, use the formula: Kcom = (m1r1 + m2r2) / (m1+m2), where m1 and m2 are the masses and r1 and r2 are the positions of the masses.
- The Com can be found by taking moments of the masses along a line.
- The Com can also be calculated continuously by integrating over the mass distribution.
Position of Center of Mass
- The position of the Com depends on your choice of origin.
- (xcm)1 = m2R / (m1+m2) and (xcm)2 = m1R / (m1+m2), where R is the distance between the two masses.
Special Bodies
- The Com of a rod is at its midpoint.
- The Com of a hay disc is at 4R/3Ï€, where R is the radius of the disc.
- The Com of a half ring is at 2R/Ï€.
- The Com of a hollow hemisphere is at 4R/8.
- The Com of a hollow core is at h/3, where h is the height of the core.
- The Com of a solid core is at h/4.
- The Com of a potential hemisphere is at 4Ï€R/8.
Cavity Wall Problems
- The Com of a cavity and the remaining mass of the object are equal.
Shift in Position of the Com
- The change in the position of the Com can be calculated using the formula: drcm = (m1dr1 + m2*dr2) / (m1+m2), where dr1 and dr2 are the changes in position of the masses.
- For no movement of the Com, m1dr1 = -m2dr2.
Acceleration of the Com
- The acceleration of the Com is given by acm = (m1a1 + m2a2) / (m1+m2).
- In a pulley block system, the acceleration of the Com is: acm = -(m2-m1)^2 / (m1+m2)*g.
Momentum
- The momentum of a body is defined as P = mv, where m is the mass and v is the velocity.
- Newton's second law states: F = dP / dt.
- If the net external force on a system is zero, the momentum of the system is conserved.
- This means that the velocity of the Com is constant, and the acceleration of the Com is zero.
- The velocity of the Com can be zero, but this does not necessarily mean that the acceleration of the Com is zero.
- For a system, the external force is equal to the rate of change of the momentum of the Com: Fext = dPcm / dt.
Internal Forces
- Internal forces within a system do not affect the motion of the Com.
- However, internal forces can change the kinetic energy of the system.
Explosion
- An explosion is a process where internal forces cause a rapid increase in kinetic energy.
- In an explosion, the net external force on the system is zero, so the momentum of the system is conserved.
- The change in kinetic energy during an explosion is equal to the energy released in the explosion: Q = Kf - Ki = 1/2 * mv^2f + 1/2 * mV^2i, where Kf is the final kinetic energy and Ki is the initial kinetic energy.
Collision
- A collision is an event where two or more objects come into contact.
- The coefficient of restitution (e) is a measure of how much kinetic energy is lost during a collision.
- e = separation / muom = (V1-V2) / (U2-U1)
Elastic Collision
- In an elastic collision, kinetic energy is conserved: KEi = KEf.
- The momentum of the system is also conserved: Pi = Pf.
- The final velocities of the objects can be calculated using the following equations:
- V1 = (m1-m2 / m1+m2) * U1 + 2m2 / m1+m2 * U2
- V2 = 2m1 / m1+m2 * U1 + (m2-m1 / m1+m2) * U2
- The change in kinetic energy in an elastic collision is zero: ΔKE = 0.
Special Cases in Elastic Collisions
- When the two masses are equal, their velocities are interchanged after the collision.
Inelastic Collisions
- In inelastic collisions, kinetic energy is not conserved: KEi ≠KEf.
- Momentum is still conserved: Pi = Pf.
- The final velocities of the objects can be calculated using the following equation:
- V1 = (m1-m2 / m1+m2) * U1 + b*U2 (m1+m2 / m1+m2)
- The change in kinetic energy in an inelastic collision is given by: ΔKE =~ 1/2 * m1*m2 / (m1+m2) * (V^2i - V^2f)
Perfectly Inelastic Collisions
- In a perfectly inelastic collision, the two objects stick together after the collision.
- Momentum is still conserved: Pi = Pf.
- Kinetic energy is not conserved: KEi ≠KEf.
- The final velocity of the two objects is given by: Vo comunes = (m1*U1 + m2 *U2) / (m1+m2).
- The change in kinetic energy is given by: ΔKE = 1/2 * (m1 * m2 / (m1+m2)) *(V0)^2
Oblique Collision
- In an oblique collision, the objects collide at an angle.
- We use the following steps to analyze oblique collisions:
- Resolve the velocities of the objects into components perpendicular and parallel to the line of impact.
- Apply the conservation of momentum along both the parallel and perpendicular directions.
- If the collision is inelastic, use the coefficient of restitution to calculate the final velocities.
- If the collision is elastic, use the principle of kinetic energy conservation in addition to the momentum conservation equations.
Studying That Suits You
Use AI to generate personalized quizzes and flashcards to suit your learning preferences.
Related Documents
Description
Test your understanding of the center of mass in physics. This quiz covers formulas, calculations, and positions of the center of mass for various shapes and bodies. Challenge yourself with questions on how to determine the center of mass and its significance in physical systems.
