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Physics Lab

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1 Physics Lab 2006: Linear Mechanics
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Physics Lab 2006: Linear Mechanics

Linear mechanics addresses physics problems in one dimension. Mechanics includes the concepts of linear motion (kinematics), Newton’s Laws and forces, momentum, and energy.

Common Variables and Quantities

The acceleration due to gravity on Earth is 9.8 $m/s^2$ . This quantity is known as g.

Mass is abbreviated m.

The force or weight of an object is $F = mg$ ; the metric unit of force is the Newton, which is one $kg m/s^2$ .

Kinematics/Linear Motion

In kinematics, there are five quantities used.

t is the length of time in which the object does something. This is generally in seconds (s).

d is the displacement of an object. Displacement is not the same as distance; displacement measures the distance relative to a certain position called the origin. The origin is at 0 meters displacement, and is often where an object starts movement. However, the origin does not necessarily need to be in the same place as the object’s initial position. The concept of displacement means that an object can move farther than its final position may indicate. Thus, if I begin at the origin, move 55 meters in the positive direction, 60 meters in the negative direction, then another 5 meters in the positive direction, the distance I have covered is 120 meters, but my displacement is 0 meters. Why? Because my final position is the same as my initial position. Displacement is generally in meters (m).

$v_i$ and $v_f$ are quantities of velocity. Velocity is different from speed in that velocity has a direction, i.e. it can be positive or negative. Generally, negative movement is considered to be when an object is moving down or to the left. Average velocity is the change in displacement over the change in time. Again, because displacement is the distance from the origin, even if an object is moving, if it ends at the same point it began then it has no velocity. $v_i$ represents velocity at the beginning of a time interval, and $v_f$ represents velocity at the end of a time interval. Velocity is generally in meters per second (m/s).

a is the acceleration of an object over a time period. Acceleration is how fast an object’s velocity is changing; average acceleration is the change in velocity over the change in time. This quantity does have direction like velocity and displacement, but the sign can mean several different things. Positive acceleration occurs in two situations. First, the object is slowing down and moving to the left, or speeding up and moving to the right. How can this be? It’s because acceleration is measuring change in velocity as well as direction. Obviously, if the object is speeding up and moving to the right, change in velocity is positive because the velocity is increasing, and it is also moving to the right (+*+=+). However, if the object is moving to the left and slowing down, change in velocity is negative and the object is moving in the negative direction. Therefore, the overall quantity is positive because -*-=+. Negative acceleration occurs in two situations as well. If the object is moving to the right and slowing down, acceleration is negative because +*-=-. If the object is moving to the left and speeding up, acceleration is negative because -*+=-. Acceleration is generally in $m/s^2$ .

Using the above quantities, there are five formulas that are often used for time intervals in which acceleration is constant.

$v_f = v_i at$
$v_f^2 = v_i^2 2ad$
$d = v_it .5at^2$
$d = v_ft - .5at^2$
$d = t(v_i v_f)/2$

Given any three of the quantities, it is possible to find any of the remaining two quantities.

Free fall is a special situation of linear motion. In this situation, the acceleration is 9.8 $m/s^2$ - or whatever g is on your planet. The formulas above can still be used because the acceleration is constant, g.

Forces and Newton’s Laws

Newton's Three Laws of motion describe the properties of forces. Forces cause objects to move or otherwise do something; it is the ability to do work.

The first law states that an object at rest tends to remain at rest, while an object moving at constant velocity tends to remain at constant velocity unless acted on by an outside force. This law describes inertia. Inertia is not a force; in fact, it is the absence of force. It describes an object's resistance to change in motion.

The second law is the basis for everything in forces. It is simply $\sum F=ma$ . Sum of all forces on an object equals object mass times object acceleration.

The third law states "For every action there is an equal yet opposite reaction." But if this is true, how could anything happen at all; if the reaction is equal and opposite, why should anything happen. It is because the action and reaction occurs on different objects. For example, I have two blocks. I push one into the other, and the two blocks move to the right. Why is this possible? Because the reaction occurs on a different object than the action.

Example Problem

The three laws are used to explain the relationships of forces within a system. Visually, this is best accomplished with a free-body diagram. To make a free body diagram, draw all of the forces that apply to an object. For a small box m1 at rest on a table, the free body diagram would look like this:

The forces on this block are that of gravity and a force normal. The force of gravity is the mass of the object times g. This is because of the Second Law $F=ma$ . The mass, $m_1$ , is pulled down at the acceleration of gravity, which is defined by g (9.8 $m/s^2$ on earth). So, $F_g = m_1g$ . If the object were in the air, it would be in free fall. The free body diagram would look almost the same, but without $F_N$ the object is pulled by gravity in the direction shown.

However, the object is not accelerating because the table is pushing back with an equivalent force. This force is the force normal ( $F_N$ ), so called because it pushes back along the normal (an imaginary line that runs straight out of a surface). In this case, $F_g = F_N$ because the box is not on a ramp and does not have any vertical forces.

$\sum F= m_1a = F_n - F_g = 0$

The sum of the forces is zero since $F_g = F_N$ . The forces are subtracted because they are acting in opposite directions. However, if there were another force acting vertically, as shown:

the equations would have to be changed, and $F_N$ would no longer equal $F_g$ . It would now have to oppose both gravity and the applied force $F_A$ down.

$\sum F= m_1a = F_n - F_g - F_A$

$a$ still equals zero, since the object is not accelerating through the table, so, by application of some basic algebra, $F_n$ can be found through: $F_n = F_g F_A$

Example Ramp

For an object on a ramp, the process is different. The free body diagram appears as shown:

The box is now at an angle, so all of the forces are now at the same angle. To make the equations easier, it helps to reorient the frame of reference so that you view $F_n$ as straight up.

So $F_n$ is familiar. $F_f$ is friction, the force that opposes another force due to rubbing along a surface. The rougher the surface, the more friction (surface area has nothing to do with it). Simply put, friction opposes movement. $\theta$ (theta) is the angle of the ramp.

The final two are force parallel and force perpendicular. These are components of mg. Forces are vectors, which are quantities with magnitude and direction that can be subdivided into smaller vectors called components. Vectors can be represented with a right triangle; the legs of the triangle are the components of the hypotenuse, which is the resultant vector. So, in reality, $mg$ has not disappeared, but has been divided into its components for simplicity:

The forces on an object on a ramp are defined as follows:

Perpendicular Force = $mgcos(\theta)$
Parallel Force = $mgsin(\theta)$
Frictional Force = $mgcos(\theta)(\mu)$

When an object slides down a ramp at constant velocity, the frictional force and parallel forces are equal.

Mu ( $\mu$ )is the constant of friction for two surfaces rubbing against each other. Mu comes in two forms, static friction and kinetic friction. When an object is at rest, the object is held at rest by the constant of static friction. This static friction must be overcome in order to move the object. At the point when the object just begins to slide (i.e. is moving at constant velocity), the force applied to the object is precisely equal to the static frictional force. As the object accelerates, the friction is kinetic friction, and is calculated by use of the constant of kinetic friction. These are two distinct constants, and must be applied in the proper situations.

For example, a block sitting motionless on a ramp is held in place by static friction. A block accelerating down the ramp is being slowed by kinetic friction.

An object with no acceleration is said to be in equilibrium. Thus, $F=ma=0$ . There are two kinds of equilibrium, static and dynamic. Static equilibrium is when there are no forces acting on the object, or the forces in the positive and negative directions are equal. In either case, the object cannot accelerate. Dynamic equilibrium is when the object is moving at constant velocity, which means acceleration is zero. Mathematically speaking, these two situations are identical. So, if the object is not accelerating, the force equations equal zero.

Since the ramp has two dimensions, two equations are necessary—one for vertical forces and one for horizontal forces (remember, we reoriented the ramp, so horizontal really means down the ramp). The equations for an object m1 accelerating down a ramp are:

$\sum F_x= m_1a = F_|_| - F_f$

$\sum F_y= m_1a = F_N - F_p_e_r_p_e_n_d_i_c_u_l_a_r$

Work and Energy

$KE = 1/2mv^2$
$GPE = mgh$

Linear Momentum

The main thing to remember for this topic is that momentum is always conserved within a system. That means that the momentum before collision of a system will always be equal to the momentum after collision of a system. For an elastic collision, the kinetic energy of the system is also conserved.

Links

New York Coaches Conference [1]

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