Chapter 04 : Newton's Laws

 

Example: Object on Incline Held in Place
An object of mass M=100 kg is placed (at rest) on a 40o ramp. We want to hold the object in place by pushing on it horizontally as shown in the figure. How much force do we need to apply? (Fpush=?) Procedure:  •  Object = ?  •  Forces acting on that object = ?  •  Coordinate system = ?  •  Apply   \( \sum \vec{F}_i = m\vec{a} \)   to that object (vectors!)

 

Tension Force FT
Rubber band : stretched Molecular bonds being stretched Restoring Force along the stretched direction Same occurs with solid materials (wire, cable, rods, beams, ...) \( \Delta l = \frac{1}{E} \frac{F}{A} l \)     (Young's modulus; PH2233) FT itself is basically a scalar and is present along the entire length of the medium that has been stretched.
NOTATIONAL CONVENTION when multiple tensions are present:
Book uses FT and then adds another subscript: not ideal when we add yet another subscript for x and y components...	\[ F_{T_1} \hspace{2em} or \hspace{2em} F_{T_A} \]
Previous book: T1 with components T1x etc. Cleaner notation, so I'll use this when multiple tensions are present.

 

Hanging Lamp
A 20 kg lamp is hanging from the ceiling from a wire.   Determine the amount of tension in the wire. What about where the wire is connected to the ceiling? What force is the wire exerting on the ceiling?

 

Hanging Lamp (revised)

Suppose the person using this room decides that the lamp is in the wrong place. They attach another wire (horizontally) that is pulling the lamp over the side so that the original wire is now making a 30o angle relative to the vertical.   What tensions are present now? What effect does this have at the ceiling?   Where the wire is connected to the ceiling, it's pulling down but always sideways. Many (most?) materials have different (weaker) strength against shear forces like this.

 

Hanging Lamp (revised again)

Suppose the person using this room decides that the lamp is in the wrong place. This time, they pull it over to the left by solidly attaching (welding?) a horizontal wire to the (originally) vertical wire, instead of to the lamp itself.   Again, suppose the original wire is now making a 30o angle relative to the vertical where it's connected to the ceiling, but becomes vertical below the 'weld' point.   What tensions are present now?

 

Airplane Acceleration

Suppose we're sitting in an airplane that's about to take off, and are holding an object hanging on a piece of string. As the plane starts accelerating down the runway (assume the acceleration is constant), we notice that the object swings towards the back of the plane and hangs there with the string making an angle of 25o relative to the vertical. How does the angle relate to the plane's acceleration? If it takes 19 sec to take off (starting at rest) what is the take-off speed? How far did the plane travel before taking off?

 

Angle vs Acceleration	Acceleration vs Angle
Cheap 'Accelerometer'