Lecture 25 - Center of Mass

center of mass (cm)

Any system (or single object) is composed of many particles, but we can consider it as a point by locating a special point in space called the object’s ___________.

center of mass (cm)

Water molecule: Has three atoms, but can be represented as a point at ___________ (x mark in the figure)

(Product of masses and X/Y)/ Sum of masses

For a system of bodies with masses m₁, m₂, …. m_n, located at r₁, r₂, … r_n, the position of the center of mass r_cm is:

geometric center

For geometrically symmetrical objects (sphere, cube, cylinder) of uniform density, the center of mass is the _____________.

(m₁ + m₂ + ….. + m_n)_Vcm = m₁v₁+ m₂v₂ + ….. + m_nv_n

Why is M_Vcm = P_total?

external forces

For a system consisting of many particles/objects, all the forces that act internally would cancel out each other due to Newton’s 3rd law. Only __________ will contribute to F_net.

a_cm = F_net / M_total

Calculate the acceleration of the center of mass: F_net = Ma_cm

m_B = (m_Ax_A-m_Ax_cm)/ (x_cm-x_B)

Calculate the mass of a particle B. (m_a+m_b)x_cm = m_Ax_A + m_Bx_B

constant

For a system of particles on which ∑F_ext = 0. The total momentum is _________, and the velocity of the center of mass is also __________.

Total Momentum / Total Mass = Velocity of Center of Mass

Puck. During hockey practice, two pucks are sliding across the ice in the same direction. At one instant, a 0.18 [kg] puck is moving at 16 [m/s] while the other puck has a mass of 0.14 [kg] and a speed of 3.8 [m/s]. What is the velocity of the center of mass of the two pucks?

Calculate for initial center of mass, plug into the second equation with the change in position to find, the missing change in position.

James (mass: 90.0 [kg]) and Ramon (mass: 60.0 [kg]) are 20.0 [m] apart on a frozen pond. Midway between them is a mug of their favorite beverage. They pull on the ends of a light rope stretched between them. When James has moved 6.0 [m] toward the mug, how far and in what direction has Ramon moved?

Find the range to get the Center of Mass, after this make sure to set the system of equations to gain the position of x₂.

A banana was thrown with an initial velocity of 7.0 [m/s], 15° from the horizontal. A second before it lands on the ground, the banana exploded, splitting the banana into two pieces. The two pieces reached the ground at the same moment. If the 0.10 [kg] piece ended up with a horizontal distance of x1 = 1.0 [m] from the launching position, where did the 0.020 [kg] piece end up?