Question

Warm up questions:

Display will be a single object resting on a board balanced on top of a pedestal; the other will be two objects resting on a board balanced. You do not know precisely what objects you will have to work with

1. Draw a picture for both mass models. Select a coordinate system. Identify and label the quantities you can measure in this problem, such as masses and lengths. The unknown quantity in this problem is the locations of the mass in relation to the balance point.

2. On your picture, identify each force on your system. Draw a free-body diagram of the board that includes distances from the balance point for each force. For now,
identify an arbitrary balance point for the system. (This is OK, you will use the condition of equilibrium to make the correct choice in a bit.)

3. Write down an expression for the net torque on the system. What is the net torque when the system is in equilibrium?

4. How many unknowns are there in your torque equation? Since we have too many unknowns to solve this equation, we need to add an additional constraint. Make the object mass twice the meter stick mass.

5. Solve the equation to find the equilibrium location of the mass.

Answer 1

1. the picture and FBD of the two scenarios is as under

3. case 1, net torque on the system about the pivot point = T1

T1 = m1*g*x - m2*g*(L/2 - x- d)

case 2

T2 = m1g*x - m2g*(L/2- x - d2) - m3*g(L/2 - x - d3)

for equilibrium, net torque = 0

4. for case 1, m2 = 2m1

for case 2, m2 = 2m1

m3 = 2m1

5. for equilibrium, T1 = T2 = 0

case 1

T1 = 0

m1*g*x = 2m1*g*(L/2 - x - d)

x = L - 2x - 2d

3x = L - 2d

d = (L - 3x)/2

for the next case

T2 = 0

m1*g*x = 2m1*g(L/2 - x - d2) + 2m1*g(L/2 - x - d3)

x = (L - 2x - 2d2) + (L - 2x - 2d3)

5x = 2L - 2(d2 + d3)

d2 + d3 = (2L - 5x)/2