1. How do we explain the motion of the center of mass of a system?

2. Locating the Center of mass in 2 dimensions-Problem 3
A system consists of three particles located at the corners of the triangle as shown below. Find the center of mass of the system.
x=(d +5/7 b)i y=(4/7 h)j

3. Locating the Center of Mass in 2 dimensions-Problem 4
A uniform piece of sheet steel is shaped. Compute the x and y coordinates of the center of mass of the piece.
11.7i+13.3j

4. Finding the Center of Mass by Integration
If the system has a continuous mass distribution, then we must using integration to find the center of mass. rcm = (1/M)∫rdm

5. Symmetry
The center of mass of a homogeneous, symmetric body must lie on the axis of symmetry.

6. Uniform Rod
Where do you think the center of mass of a uniform rod is? We can write an equation for the linear mass density, λ = dm/dx. So dm = λdx AND xcm = 1/M ∫xdm

7. Finding the Center of Mass of a Non-Uniform Rod
A rod of length 30.0 cm has a linear density (mass per length) given by λ=50.0g/m + (20.0g/m2)x
Where x is the distance from one end, measured in meters.
What is the mass of the rod?
How far from the x = 0 end is its center of mass?
M=15.9 g
x = cm

8. Finding the Center of Mass of a Semicircular Hoop
Write down equation for center of mass. Write down an expression for dm. Write down an expression for x. Write down an expression for y.
dm=λdl = λrdθ

9. Center of Mass of Semicircular Hoop
rcm= 1/M∫(rcosϴi +rsinϴj)dm Substitute for dm and we get: rcm = r2/M ∫(cosϴi +sinϴj)λdθ The mass density… λ=M/πr Show that rcm=2r/πj

10. Motion of the Center of mass
The velocity of the center of mass can be expressed as: vcm = (1/M)∑mivi (Derive this) If there is no external forces acting on the system, what must be true of the velocity of the center of mass?

11. Calculating the velocity of the center of mass
A 5 kg particle moves along the x-axis with a velocity of 3.0 m/s. A 2.0 kg particle moves along the x axis with a velocity of -2.5 m/s. Find the velocity of the center of mass.
1.4 m/s to the right

12. Thought Question on Velocity of Center of Mass
The four particles shown below all have equal mass. Their velocities and positions are indicated as well. The magnitudes of their velocities are equal.
Which pairs of particles form a center of mass which is stationary?
Which pairs of particles form a center of mass which is stationary and at the origin?
Which pairs of particles form a system with a center of mass that passes through the origin?

14. How does Momentum Relate to Velocity of Center of Mass?
Consider the system below
Calculate the total momentum of the system.
Calculate the velocity of the center of mass of the system.
Multiply the velocity of the center of mass by the total mass of the system.

15. How does Momentum Relate to Velocity of Center of Mass?
The total momentum of a system is equal to the velocity of the center of mass multiplied by the total mass of the system. (If the total momentum of a system is 0, does the center of mass move?)

16. Thought Question-2D Collision and Center of Mass
Two bodies that form a closed, isolated system undergo a collision along a frictionless floor. Which of the three choices best represents the path of those bodies and the path of their center of mass as seen from overhead?

17. Canoe
A boy stands at one end of a canoe that is stationary relative to the shore. He then walks to the opposite end of the canoe, away from the shore. Does the canoe move?

18. Velocity of Center of Mass- Challenge
Romeo (77 kg) entertains Juliet (55.0 kg) by playing his guitar from the rear of their boat at rest in still water, 2.70 m away from Juliet who is in the front of the boat. After the serenade, Juliet carefully moves to the rear of the boat (away from shore) to plant a kiss on Romeo’s cheek. How far does the 80 kg boat move toward the shore it is facing?
0.70 m

19. Problem-Velocity of Center of Mass
A man of mass m stands at the rear of a uniform plank of equal mass and length L that rests on a frictionless surface. The man walks to the center of the plank. Relative to the ice, the man has moved a) L/2 b) L c) L/4 d) 3L/4 e) L/3

20. Explosion
A projectile is fired into the air and suddenly explodes into several fragments. What can be said about the motion of the center of mass of the fragments after the explosion?

21. Why?

22. Acceleration of the Center of mass
acm = (1/M)∑aimi (Derive this) The center of mass of a system moves like a particle of mass M = ∑mi under the influence of the net external force acting on a system.