Center of Mass and Newton's Second Law

PHYS 211

Center of mass of point masses?

$$x,com=\frac{\sum_{}^{}mixi}{\sum totalmass}$$

Multiply each mass by its position, add, then divide by total mass.

Center of mass in 2D?

Find x_com and y_com separately using the mass-weighted average.

Center of mass on an L-shaped plate?

Treat the missing piece as negative mass:

$$x,com=\frac{M1\cdot x1-M2\cdot x2}{M1-M2}$$

Repeat for y.

How do you find mass from area?

For a uniform thin plate,

$$m=\sigma A$$

[σ = surface mass density (kg/m²)]

If density is constant, areas can replace masses in the COM formula.

Newton’s 2nd law for a system?

$$\sum_{}^{}\overrightarrow{F},ext=M\overrightarrow{a,}com$$

Add external forces, then divide by total mass.

How do you handle forces in x and y?

Solve separately:

$$a,x=\frac{\sum_{}^{}F,x}{M}$$

$$a,y=\frac{\sum_{}^{}F,y}{M}$$

Keep positive and negative signs.