CAPM Challenge I

Challenge: Find the acceleration of a ball rolling down a table using only a ball, chalk, and ruler. Also predict the velocity of the ball at 4 seconds. 

With the data recorded, we calculated the velocity and acceleration of the ball. We used the data to create a graph and calculated the average velocity using the formula: V = ΔD/ΔT.

V = (123.3-0)/(4-0) or (123.3/4), which equals 30.825 cm/s

The average velocity of the ball during the trip was calculated to be .30825 meters/s^2.

We had to predict the position of the ball at 4.5 seconds in order to calculate the ball's velocity at 4 seconds. We estimated the position of the ball would be 148.4 centimeters. So with this information, we calculated the velocity of the ball at 4 seconds. 

(148.4 - 100.2)/(4.5  - 3.5) or (148.4 - 100.2)/(1) = 48.2 cm/s or .482 m/s²

The acceleration can be calculated using the formula: A = ΔV/ΔT.

A = (30.825 - 0)/(4 - 0) = 7.7 cm/s or .077 m/s²

Conclusion: The rate at which the ball travels down the table is dependent upon how tilted the surface (table) is; the greater the tilt of the surface (table), the faster the rate at which the ball will travel down it. The ball accelerates down the table because of an unbalanced force acting upon it. The normal force acts in a direction perpendicular to the surface. The ball's velocity increases as it travels down the table, as seen by the Position vs Time graph. Since the of the slope of the P vs T graph is positive, the acceleration will also be positive. The acceleration was calculated to be .482 m/s².

BFPM Practicum

Objective: Determine the weight of an object hanging from two objects at unequal angles to each other. 

Step 1: Collect known data. We know the tension of each rope (FT1 = 2.2 Newtons, and FT2 = 1 N) and angle measurements.
Step 2: Draw Free-Body Diagram of the situation (include quantities).
Step 3: If a force's vector does not align with the axes, than add a vector for each force so that it does align. So create FTx1 and FTx2, FTy1 and FTy2.
Step 4: Angle Measurements: ∠FT1 and FTy = 20º, ∠FT1 and FTx = 70º, ∠FT2 and FTy = 55º, ∠FT2 and FTx = 35º
Step 5: Use Cosineθ = Adj/Hyp which FT1 is the Hypotenuse and FTy is the Adjacent for the first part. For the second part of the problem, Use Cosineθ = Adj/Hyp which FT2 is the Hypotenuse and FTy2 is the Adjacent.
Step 6: Plug in the measurements into one of the formulas. ∠FT1 and FTy = 20º, so Cos(20º)=Adj/2.2. ∠FT2 and FTy = 55º, so Cos(55º) = Adj/1. 
Step 7: Cos(20º)=Adj/2.2. To calculate this part of the weight, just multiply Cos(20º) with 2.2 and you should calculate a number close to 2.067 Newtons.
Step 8: Cos(55º) = Adj/1. To calculate this part of the weight, just multiply Cos(55º) with 1 and you should calculate a number close to .573 Newtons.
Step 9: To calculate the weight of the object, just add the two quantities (2.067 Newtons and .573 Newtons) and you should calculate a number close to 2.64 Newtons.

Unit 2 Summary

• In this unit, I learned about Newton's First and Third of Motion and applied them to a multitude of scenarios.  We also learned about how forces act upon objects and the Balanced Force Model.

Newton’s First Law: The first law says that an object at rest tends to stay at rest, and an object in motion tends to stay in motion, with the same direction and speed.

Newton’s Third Law: The third law says that for every action there is an equal and opposite reaction. In addition, forces are found in pairs.

Fg = 10N/kg (mass)

How do seat belts keep you safe?

Newton’s First Law of Motion says that unless an outside force acts on an object, the object will continue to move at its present speed and direction. Unless the objects inside the car are restrained they will continue moving at whatever speed the car is traveling even if the car is stopped by a crash. If the passenger is restrained by a seat-belt, their momentum is reduced more gradually by the constant and smaller force of the belt acting over a longer period of time.

Newton's First Law Applied: 

The soccer ball will remain at rest unless acted upon by an external force. The foot applies a push force to the ball and the ball will continue with constant speed and direction unless acted upon by an unbalanced force. 

To examine the forces that are being acted upon an object, we use a free-body diagram. A free-body diagram is a sketch of the object that shows all of the forces acting on the object shown.    

In this free-body diagram, we are examining the forces acting upon a football that has has been kicked into the air. The football was at rest until an unbalanced force acted upon it (the foot). As the football travels at a constant speed in the air, a normal force and a gravitational force act upon the football. The football will continue to travel at a constant speed until an unbalanced force acts upon it. 

What exactly are balanced and unbalanced forces?

• Forces occur in pairs and can be either balanced or unbalanced. Balanced forces do not cause a change in motion. They are equal in size and opposite in direction.
• Unlike balanced forces, unbalanced forces always cause a change in motion. They are not equal and opposite.

Newton's Third Law Applied: 

If someone pushes horizontally against a wall with a force of 100 N, then the wall will push horizontally against the person with a force of 100 N. 

To examine the forces that are being acted upon an object, we use a free-body diagram. 

In this free-body diagram, we are examining the forces acting upon the ball and the wall. The ball's push against the wall is equal to the wall's push force on the ball. There will always be a reaction-action pair. The ball pushes forward on the wall; the wall pushes back on the ball with an equal force. We know this because of Newton's Third Law: for every action, there is an equal and opposite reaction. 

In this example, two forces act upon the motionless cat: normal force and gravitational force. The floor provides the normal force and the earth's gravity provides the gravitational force. These forces are balanced, so the cat remains at rest.

What about objects suspended by the ceiling? 

In this example, two forces act upon this chandelier suspended by the ceiling: tension force and gravitational force. The chain attached to the ceiling provides the tension force and the earth's gravity provides the gravitational force. These forces are balanced, so the chandelier remains at rest. 

What are force diagrams?

• A force diagram is simply a diagram showing all the forces acting on an object, the force's direction and its magnitude.

Some examples: 

In each example there is a net force acting upon the object. The net force is the vector sum of all the forces that act upon an object. That is to say, the net force is the sum of all the forces, taking into account the fact that a force is a vector and two forces of equal magnitude and opposite direction will cancel each other out.    

In the examples above, the net force is determined by summing the individual force vectors that are acting upon the objects.    

INCLINED PLANES:

A box can be moved up an inclined plane with constant velocity by a push force of magnitude F1 or down the inclined plane with constant velocity by a force of magnitude F2. When creating a Force Diagram representing the forces acting upon the box, the axes should be aligned parallel and perpendicular to the hill. Rotate the coordinate axis and it will make analyzing much easier.

The box in both cases is affected by these forces:

F⃗ G - the weight of the box

F⃗ 1(F⃗ 2)- force pushing the box

N⃗  - force by which the inclined plane affects the box (the normal force)

F⃗ t - Friction force

What is the coefficient of kinetic friction?

μk: Symbol

μk   = F_k / N  (F_{k = Force of kinetic friction), (N =} Normal force or the force perpendicular to the contacting surfaces)

The motion equation for the box that moves up the hill:

F⃗ G+N⃗ +F⃗ 1+F  = 0⃗ t

The motion equation for the box that moves down the hill:

F⃗ G+N⃗ +F⃗ 2+F⃗ t=0

Force Diagrams and Component Forces: Using SOHCAHTOA

SOH stands for Sine equals Opposite over Hypotenuse. CAH stands for Cosine equals Adjacent over Hypotenuse. TOA stands for Tangent equals Opposite over Adjacent.

sinθ = opp/adj, cosθ = adj/hyp, tanθ = opp/adj

Vector: A quantity possessing both magnitude and direction, represented by an arrow the direction of which indicates the direction of the quantity and the length of which is proportional to the magnitude.

In this example, need to calculate the tension in the cable that is supporting this object. The equation for the forces in the Horizontal (x) Direction: FTBx + FTAx = 0 N and the equation for the forces in the Vertical (y) Direction: Fg + FTBy + FTAy = 0 N. We know the weight of the object is equal to 25 N and that the cables make a 30 degree angle with the object. FTy = 12.5 N so we would use Sin(30°) = 12.5/FT1. We would calculate this and FT2 should equal 25 N.  The object exerts 25 N on the cable and the cable exerts 25 N on the object, representing a reaction-action pair! So the forces acting upon this object are equal, therefore the object will remain at rest unless an unbalanced force acts upon it. 

)00