1. Moments and Centers of Mass
Section 7.6a

2. See-Saw

3. Moments Moment = 𝑚𝑥 Moment about the origin
𝑀 𝑜 = 𝑚 1 𝑥 1 + 𝑚 2 𝑥 2 + 𝑚 3 𝑥 3 +…+ 𝑚 𝑛 𝑥 𝑛
Moments

4. Moments 𝑚 1 =6 𝑘𝑔 𝑚 2 =10 𝑘𝑔 𝑚 3 =20 𝑘𝑔 𝑥 1 =1 𝑚 𝑥 2 =5 𝑚 𝑥 3 =−6 𝑚
Calculate the moment about the origin for the system. Is the system in equilibrium? If not, which way will it tilt?
𝑚 1 =6 𝑘𝑔 𝑚 2 =10 𝑘𝑔 𝑚 3 =20 𝑘𝑔
𝑥 1 =1 𝑚 𝑥 2 =5 𝑚 𝑥 3 =−6 𝑚
𝑚 1 =5 𝑘𝑔 𝑚 2 =9 𝑘𝑔 𝑚 3 =12 𝑘𝑔
𝑥 1 =0 𝑚 𝑥 2 =8 𝑚 𝑥 3 =−6 𝑚
𝑚 1 =50 𝑘𝑔 𝑚 2 =3 𝑘𝑔 𝑚 3 =14 𝑘𝑔 𝑚 4 =30 𝑘𝑔
𝑥 1 =4 𝑚 𝑥 2 =−8 𝑚 𝑥 3 =−10 𝑚 𝑥 4 =0 𝑚

5. A 70 kg person is sitting 10 feet from the fulcrum of a see-saw
A 70 kg person is sitting 10 feet from the fulcrum of a see-saw. A 50 kg person is sitting 10 feet in the other direction. Where should the fulcrum be placed in order to balance the see-saw?
Balancing Point

6. 𝑥 = 𝑀 𝑜 𝑚 Center of Mass 𝑚 1 =6 𝑘𝑔 𝑚 2 =10 𝑘𝑔 𝑚 3 =20 𝑘𝑔
𝑥 = 𝑀 𝑜 𝑚
Find the center of mass of the system from the previous examples:
𝑚 1 =6 𝑘𝑔 𝑚 2 =10 𝑘𝑔 𝑚 3 =20 𝑘𝑔
𝑥 1 =1 𝑚 𝑥 2 =5 𝑚 𝑥 3 =−6 𝑚
𝑚 1 =5 𝑘𝑔 𝑚 2 =9 𝑘𝑔 𝑚 3 =12 𝑘𝑔
𝑥 1 =0 𝑚 𝑥 2 =8 𝑚 𝑥 3 =−6 𝑚
𝑚 1 =50 𝑘𝑔 𝑚 2 =3 𝑘𝑔 𝑚 3 =14 𝑘𝑔 𝑚 4 =30 𝑘𝑔
𝑥 1 =4 𝑚 𝑥 2 =−8 𝑚 𝑥 3 =−10 𝑚 𝑥 4 =0 𝑚

7. 2-Dimensional Systems 𝑥 = 𝑀 𝑦 𝑚 and 𝑦 = 𝑀 𝑥 𝑚
Find the center of mass of a system of point masses 𝑚 1 = 𝑚 2 = 𝑚 3 =2, and 𝑚 4 =9, located at (3, -2) (0,0) (-5, 3) (4, 2)
𝑥 = 𝑀 𝑦 𝑚 and 𝑦 = 𝑀 𝑥 𝑚
𝑀 𝑥 = 𝑚 1 𝑦 1 + 𝑚 2 𝑦 2 + 𝑚 3 𝑦 3 +…+ 𝑚 𝑛 𝑦 𝑛
𝑀 𝑦 = 𝑚 1 𝑥 1 + 𝑚 2 𝑥 2 + 𝑚 3 𝑥 3 +…+ 𝑚 𝑛 𝑥 𝑛

8. 2-Dimensional Systems
Find the center of mass of a system of point masses 𝑚 1 = 𝑚 2 =5 𝑚 3 =3, and 𝑚 4 =10, located at (5, 0) (1,8) (-0, -2) (5, -4)