1. Physics 106: Mechanics Lecture 08
Wenda Cao
NJIT Physics Department

2. Static Equilibrium Important announcement
Equilibrium and static equilibrium
Static equilibrium conditions
Net external force must equal zero
Net external torque must equal zero
Solving static equilibrium problems
March 10, 2011

3. Static and Dynamic Equilibrium
Equilibrium implies the object is at rest (static) or its center of mass moves with a constant velocity (dynamic)
106 deals only with the special case in which linear and angular velocities are equal to zero, called “static equilibrium” : vCM = 0 and w = 0
Examples
Book on table
Puck sliding on ice in a constant velocity
Ceiling fan – off
Ceiling fan – on
Ladder leaning against wall (foot in groove)
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4. Conditions for Equilibrium
The first condition of equilibrium is a statement of translational equilibrium
The net external force on the object must equal zero
It states that the translational acceleration of the object’s center of mass must be zero
March 10, 2011

5. Conditions for Equilibrium
The second condition of equilibrium is a statement of rotational equilibrium
The net external torque on the object must equal zero
It states the angular acceleration of the object to be zero
This must be true for any axis of rotation
March 10, 2011

6. Conditions for Equilibrium
The net force equals zero
If the object is modeled as a particle, then this is the only condition that must be satisfied
The net torque equals zero
This is needed if the object cannot be modeled as a particle
These conditions describe the rigid objects in equilibrium analysis model
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7. Equilibrium Equations
We will restrict the applications to situations in which all the forces lie in the xy plane
There are three resulting equations
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8. B) Find where the father should sit to balance the system at rest.
A) Find the magnitude of the upward force n exerted by the support on the board.
B) Find where the father should sit to balance the system at rest.
March 10, 2011

9. Axis of Rotation
The net torque is about an axis through any point in the xy plane
Does it matter which axis you choose for calculating torques?
NO. The choice of an axis is arbitrary
If an object is in translational equilibrium and the net torque is zero about one axis, then the net torque must be zero about any other axis
We should be smart to choose a rotation axis to simplify problems
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10. P O B) Find where the father should sit to balance the system at rest.
Rotation axis O
Rotation axis P P O
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11. Center of Gravity
The torque due to the gravitational force on an object of mass M is the force Mg acting at the center of gravity of the object
If g is uniform over the object, then the center of gravity of the object coincides with its center of mass
If the object is homogeneous and symmetrical, the center of gravity coincides with its geometric center
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12. Problem-Solving Strategy 1
Draw sketch, decide what is in or out the system
Draw a free body diagram (FBD)
Show and label all external forces acting on the object
Indicate the locations of all the forces
Establish a convenient coordinate system
Find the components of the forces along the two axes
Apply the first condition for equilibrium
Be careful of signs
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13. Horizontal Beam Example
A uniform horizontal beam with a length of l = 8.00 m and a weight of Wb = 200 N is attached to a wall by a pin connection. Its far end is supported by a cable that makes an angle of  = 53 with the beam. A person of weight Wp = 600 N stands a distance d = 2.00 m from the wall. Find the tension in the cable as well as the magnitude and direction of the force exerted by the wall on the beam.
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14. Horizontal Beam Example, 2
Analyze
Draw a free body diagram
Use the pivot in the problem (at the wall) as the pivot
This will generally be easiest
Note there are three unknowns (T, R, q)
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15. Horizontal Beam Example, 3
The forces can be resolved into components in the free body diagram
Apply the two conditions of equilibrium to obtain three equations
Solve for the unknowns
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16. Problem-Solving Strategy 2
Choose a convenient axis for calculating the net torque on the object
Remember the choice of the axis is arbitrary
Choose an origin that simplifies the calculations as much as possible
A force that acts along a line passing through the origin produces a zero torque
Be careful of sign with respect to rotational axis
positive if force tends to rotate object in CCW
negative if force tends to rotate object in CW
zero if force is on the rotational axis
Apply the second condition for equilibrium
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17. Horizontal Beam Example, 3
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18. Problem-Solving Strategy 3
The two conditions of equilibrium will give a system of equations
Solve the equations simultaneously
Make sure your results are consistent with your free body diagram
If the solution gives a negative for a force, it is in the opposite direction to what you drew in the free body diagram
Check your results to confirm
March 10, 2011

19. Example 3: What do the scales read?
A uniform beam, of length L and mass m = 1.8 kg, is at rest with its ends on two scales (see figure). A uniform block, with mass M = kg, is at rest on the beam, with its center a distance L / 4 from the beam's left end.
What do the scales read?
Solve as equilibrium system
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20. 1: 2:
3: Choose axis at O O
From 2:
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21. Example 4:
A safe whose mass is M = 430 kg is hanging by a rope from a boom with dimensions a = 1.9 m and b = 2.5 m. The boom consists of a hinged beam and a horizontal cable that connects the beam to a wall. The uniform beam has a mass m of 85 kg; the mass of the cable and rope are negligible.
What is the tension Tc in the cable; i. e., what is the magnitude of the force Tc on the beam from the horizontal cable?
What is the force at the hinge? Tr Tc mg Fh Fv
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22. Choose axis at hinge
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23. Example 7: Rock Climber m2 m1
In the figure, a rock climber with mass m = 55 kg rests during a “chimney climb,” pressing only with her shoulders and feet against the walls of a fissure of width w = 1.0 m. Her center of mass is a horizontal distance d = m from the wall against which her shoulders are pressed. The coefficient of static friction between her shoes and the wall is m1 = 1.1, and between her shoulders and the wall it is m2 = To rest, the climber wants to minimize her horizontal push on the walls. The minimum occurs when her feet and her shoulders are both on the verge of sliding. m1 m2
What is the minimum horizontal push on the walls?
What should the vertical distance h be between the shoulders and feet, in order for her to remain stationary?
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24. What is the minimum horizontal push on the walls?
What should the vertical distance h be between the shoulders and feet, in order for her to remain stationary? O
Choose axis at “O”
w = 1.0 m
March 10, 2011