SAT Physics Dynamics - Inertia

SAT Physics Dynamics - Inertia

INERTIA
Inertia is the tendency of an object to resist changes in its natural motion. Galileo suggested that the natural state of motion of an object is constant velocity. Even an object at rest has a constant velocity of zero. Inertia is the tendency of a stationary object to remain at rest and for a moving object to continue moving at constant velocity. Inertia is a property of mass. The greater the mass of an object, the greater the inertia possessed by the object. Simply put, the more mass the object has, the harder it is to push around the object. Physicists regard mass as a quantity of inertia and a resistance to change in motion.

FORCE
An object cannot simply accelerate and change velocity on its own. In order to accelerate, an object must be pushed or pulled by some external source known as an agent. The push or pull of an agent that accelerates an object is known as a force. Forces are vector quantities measured in newtons.
Since forces are vector quantities, they can be added together to find a resultant sum. The sum of all forces, , acting on an object is known as the net force. The direction of the net force is the same as the direction of the acceleration (change in velocity) of the object. The relationship between the direction of force and the direction of initial velocity of an object dictates the resulting change in motion as shown in Table 5.2.

Table 5.2 Relationship Between Direction of Force and Direction of Initial Velocity

Force problems usually focus on finding the magnitude of the net force and acceleration. To make them easier to solve, set the direction of the initial velocity as positive. To make these problems easier to solve, set the direction of the initial velocity as positive. Under these conditions, forces acting in the same direction as velocity create positive acceleration and attempt to increase the speed of objects. Forces acting in the opposite direction of velocity result in negative acceleration and attempt to slow down objects.
Setting the direction of initial motion as a positive value creates a dilemma for objects that are moving downward due to gravity. Although down is technically the negative direction, speed is increasing. When solving for the magnitude (absolute value) of force or acceleration, for a falling object, it is easier to set down as the positive direction. Which direction is set as positive does not really matter. What matters is that the signs on all vector quantities match the convention chosen and remain consistent throughout the entire problem.
As a matter of convention, both one-dimensional force vectors and components of force vectors will be indicated in italics.

COMMON FORCES
Numerous agents, each with unique characteristics, are capable of generating forces. As a result, several important forces have been given unique variables.
 
Applied Force
The general variable letter, F, is used to represent any force that does not have its own vari­able designation. As an example, there is no specific variable for the force of a person pushing a box.
 
Force of Gravity (Weight)
The force of gravity, F_g, is present between any two masses. The gravity field, g, of the agent pulls on the mass, m, of the object.

Most problems take place on Earth. Unless stated otherwise, Earth is the agent creating the gravity field and g= 10 m/s². As shown in Figure 5.1, the force of gravity pulls the object with mass m toward the agent, Earth.

Figure 5.1. Force of gravity (weight)

The force of gravity is also the weight of an object. Some instructors, texts, and exams may use the variable w to represent weight (force of gravity).

Normal Force
The normal force, N, is present whenever an object pushes on a surface. There is no spe­cific formula to solve for the normal force. Rather, the normal force is a response force to an applied force. As will be shown shortly, the normal force is a result of Newton’s third law. As shown in Figure 5.2, the normal force always acts perpendicular to a surface.

Figure 5.2. Normal force

Friction
The force of friction, f or F_f, is present when two conditions are met. First, an object must be pressed against a rough surface. Second, either a force or a component of force must be act­ing parallel to the surface. Figure 5.3 shows the applied force—F, the force due to friction—f, the force due to gravity—F_g, and the normal force—N acting on an object with mass m.

Figure 5.3. Free-body diagram

In Figure 5.3, the force of gravity, F_g, pulls the object onto the surface. This causes the surface to press back with the normal force N. When any force, in this case force F,is applied paral­lel to a surface, that force will attempt to accelerate the object. If the surface is rough, then a frictional force, f, will act opposite the forward force.
Friction is described by the following equation and is dependent on two quantities:

f = μN

1. The coefficient of friction, μ (no units), indicates surface roughness.
2. The normal force, N, indicates how hard the surfaces are pressed together.

You must note that friction does not depend on surface area.
In beginning physics, friction is assumed to be absent unless its presence is indicated. Problems involving friction usually either mention a rough surface or specifically use the word “friction.” However, there is one tricky exception. If an object experiences a pushing force along a surface but does not accelerate (remains stationary or moves at constant veloc­ity), friction must be present. In these problems, friction is equal to any force attempting to move the object forward but acts in the opposite direction of any forward forces.

f = F_forward

Unfortunately, friction is one of the most complicated forces to understand in beginning physics courses. There are two types of friction. The first, static friction—f_s, occurs when the pushing force is not strong enough to overcome the friction force. As a result, the object remains static (at rest). The second, kinetic friction—f_k, is the friction of moving objects. Subscripts may also appear behind the coefficients of friction to distinguish them as either static, μ_s, or kinetic, μ_k. Remember that static friction is stronger than kinetic friction. In order to initiate motion in an object, inertia must be overcome to start the movement. Once in motion, however, the object’s forward movement and momentum now make it is easier to keep the object in motion.

Tension
Tension, T, is a force exerted by ropes and strings. There is no specific formula for tension. As with the normal force, tension is a response force and obeys Newton’s third law. Tension acts along ropes or strings as shown in Figure 5.4.

Figure 5.4 Tension

Tension has the same magnitude in every part of a particular rope or string.
 
Springs
When springs are stretched or compressed by an outside agent, a force is created in the spring. The force of a spring is known as a restoring force, and it acts to restore the spring to its original rest length. As a result, the force of a spring, F_s, always opposes the action of the agent. If an agent stretches a spring, the restoring force acts to compress the spring. If an agent compresses a spring, the restoring forces acts to stretch the spring. See Figure 5.5.

Figure 5.5. The force of a spring

The magnitude of the force of a spring is described in Hooke’s law:

The variable k is the spring constant (units: N/m), and the variable x is the distance the spring is either stretched or compressed (units: m). Every spring has its own unique spring constant. Finding the spring constant, if it is not given, is often the first critical step in solving spring problems.

FORCE DIAGRAMS
The first step in solving a dynamics problem is to identify all forces, including the direction of each force, acting on an object. This can be accomplished by constructing a force diagram, known as a free-body diagram. A free-body diagram includes only the object and the forces acting on the object. It is free of clutter and does not include surfaces, strings, springs, or any other agents acting on the object. A free-body diagram serves two key purposes: First, it allows you to visualize how the forces will influence the motion. Second, it provides a frame of reference in which to work with the force vectors.

Free-Body Diagrams

A 10-kilogram mass is pulled at constant velocity to the right along a rough horizontal surface by a string. Construct a free-body diagram depicting all the forces acting on the mass.
 
WHAT'S THE TRICK?

Identify all the forces acting on the mass.

• Gravity is not mentioned in the problem. Unless specified otherwise, problems take place on Earth, so the force of gravity, F_g, is present.
• You may be able to deduce some forces from the diagram. Both a string and a surface are in contact with the object. Therefore, tension, T, and the normal force, N, are present.
• Identifying other forces requires you to read the text of the problem. This problem mentions a “rough” surface, implying that friction, f, is present.

The resulting free-body diagram is shown below left. Free-body diagrams are nearly identical to plotting vectors on coordinate axes, as shown below right.

Including all force vectors and ensuring that they point in the correct direction are the most important aspects when drawing a free-body diagram.