PHYS101 Study Guide

Unit 4: Dynamics

4a. Compare and contrast mass and inertia

Define mass and inertia.
State Newton's First Law of Motion or the Law of Inertia.

We define mass as the amount of matter in an object. We measure mass in units such as grams. Mass does not depend on gravity and therefore does not depend on the location where it is being measured. Inertia is the property that describes the fact that an object at rest (not moving) will stay at rest unless an outside force acts upon it. Likewise, an object in motion will stay in motion with constant velocity unless an outside force acts upon it.

Newton's First Law of Motion is also called the Law of Inertia. This law states the definition of inertia: An object at rest will remain at rest unless an outside force acts upon it. Also, an object in motion with constant velocity will remain in motion with constant velocity unless an outside force acts upon it.

4b. Determine the net force on an object

Define external force and system.
Explain how acceleration changes motion.
State Newton's Second Law of Motion.
What is the unit for force? What is the meaning of this unit?

The system is whatever we are interested in when calculating a physics problem. The external force is any force that acts upon the system, but is not part of the system. For example, picture pushing a rock up a hill. The system is the rock, and the external force is you pushing the rock.

As we saw in Newton's First Law of Motion, an object at rest stays at rest unless acted upon by an external force. Also, an object in motion at constant velocity remains in motion unless acted upon by an external force. This is inertia. The only way to overcome inertia is to accelerate the object. Applying a net force to the system to induce acceleration.

Acceleration is proportional to the net external force on a system. That is, the higher the applied force, the bigger the acceleration. We also know that acceleration is inversely proportional to mass. That is, large objects accelerate at a slower rate than smaller objects. We know this from our everyday observations. It is easier to accelerate a light ball than a heavier bowling ball.

Newton's Second Law of Motion relates net external force to acceleration and mass of the system: $F_\mathrm{net} = ma$ , where $F_\mathrm{net}$ is the net force, $m$ is mass, and $a$ is acceleration.

Review a derivation of the law in the box Newton's Second Law of Motion.

The unit for force is the Newton, N. The definition of the Newton is 1 N = 1 kg m/s².

Review examples of using Newton's Second Law of Motion to calculate acceleration and force in objects in motion in Example 4.1: What Acceleration Can a Person Produce When Pushing a Lawn Mower? and Example 4.2: What Rocket Thrust Accelerates This Sled?.

4c. Draw and interpret free-body diagrams representing the forces on an object

Describe the structure of a free body diagram.
For a given system and force, draw a free body diagram.

A free body diagram shows all the forces acting upon a system within a given coordinate system. It is a simplified way to visualize what is happening in a physics problem. We draw the forces as vector arrows in the direction of the force from the center of mass of the system. This can help us to figure out how we need to add or subtract force vectors when determining the net force on an object.

Review examples of free body diagrams drawn for specific examples in Figures 4.5 and 4.6.

Review more examples of how to draw a free body diagram for a given example in Example 4.1: What Acceleration Can a Person Produce When Pushing a Lawn Mower? and Example 4.2: What Rocket Thrust Accelerates This Sled?.

4d. Identify the correct use of normal and tension forces in terms of Newton's Third Law of Motion

State Newton's Third Law of Motion or Law of Action and Reaction.
For a given system and applied force, determine the opposing force.
Define weight, normal force, tension.

Newton's Third Law of Motion states that for every force exerted by an object, there is an equal magnitude force exerted on that object in the opposite direction. This is often called the Law of Action and Reaction. That is, for every action (exerted force), there must be an equal and opposite reaction. This law tells us that forces are always paired.

Review an example of how we can apply Newton's Third Law of Motion to a swimmer in a pool in Figure 4.9. When the swimmer kicks off the wall of the pool to begin swimming, they exert a force toward the wall. Because of the Third Law, the wall also exerts an opposing force on the swimmer. The force of the wall on the swimmer is equal in magnitude, but opposite in direction of the force exerted by the swimmer on the wall. Because the force of the swimmer's feet on the wall does not exert a force on the swimmer themself, this force does not impact the swimmer. Gravity exerts a force toward the earth on the swimmer, but buoyancy exerts an equal magnitude force away from the earth. This keeps the swimmer floating in the water.

Review another example of determining the forces in a given system in Example 4.3: Getting Up to Speed: Choosing the Correct System.

We define weight as the force of gravity on an object of a given mass. Because it comes from gravity, the weight force is directed toward the earth. Now, consider a coffee cup sitting on a table. What keeps the table from collapsing under the weight force of the cup? The coffee cup is experiencing the weight force in the direction of the earth. This "pushes" down on the table. Because of Newton's Third Law of Motion, there must be an equal magnitude force in the opposite direction also acting on this table to balance the forces.

The opposing force is the normal force. Here, "normal" means perpendicular. That is, the normal force is perpendicular to the surface of the table. It balances the weight force from the coffee cup and keeps the table from collapsing. The normal force is abbreviated as N.

Review an example of how the normal force works when an object is placed on a table in Figure 4.12.

Tension is the force along the length of an object. We normally think of tension as a force in an object, such as a rope or string. Objects, such as ropes, can only exert forces in the same direction as their length. If a rope is attached to a hanging object, the object the weight of the object exerts a force toward the earth. The tension of the rope acts as a normal force in the opposite direction of the weight. Review an example of tension in Figure 4.15.

4e. Use Newton's Second Law of Motion to analyze dynamic problems

Identify forces in x and y directions for a given system.
Use Newton's Second Law of Motion to solve problems.

Whenever a problem involves forces, we must use Newton's Second Law of Motion to solve the problem. There are four steps to solving these types of problems:

Sketch the system described in the problem.
Identify forces and draw the forces on the sketch.
Draw a free body diagram of the forces acting on the system.
Use Newton's Second Law of Motion to solve the problem.

Review worked examples of solving dynamics problems using this problem-solving strategy in Example 4.7: Drag Force on a Barge, Example 4.8: Different Tensions at Different Angles, and Example 4.9: What Does the Bathroom Scale Read in an Elevator?.

4f. Give examples of the effects of friction on the motion of an object

Define friction, kinetic friction, and static friction.
Give examples of how friction impacts an object's motion.

Friction is the force between two surfaces that opposes motion between them. Kinetic friction is the friction between adjacent surfaces that are moving relative to each other. Static friction is the friction that occurs when two adjacent surfaces are not in motion. Static friction is generally higher than kinetic friction.

There are many examples of the action of friction in our everyday lives. For example, if you slide a box across a room, the box's motion will eventually stop due to the friction that occurs between the surface of the box and the surface of the floor. A box will slide relatively well across a smooth tile floor because the smooth tile floor has lower friction. It will slide less well across a floor with a rough carpet because the carpet has higher friction.

When we walk on a sidewalk our shoes do not generally slip because the friction between our shoes and the sidewalk opposes the forward force of our shoes. However, we know that icy surfaces are "slippery" when the ice exerts less friction on our shoes than concrete.

4g. State Hooke's Law

State Hooke's Law.

Hooke's Law describes oscillations or vibrations. Figure 16.1 shows oscillatory motion. When an object is deformed, there is a restoring force in the opposite direction as the deformation that works to bring the object back to its original position. Hooke's Law describes the forces of this type of motion with the following equation:

$F=-kx$

where $F$ is the restoring force, $k$ is a force constant, and $x$ is the displacement from the equilibrium position of the system.

The force constant describes how stiff the system is, or how difficult it is to deform the system.

4h. Solve problems involving springs

Identify the parts of Hooke's Law ( $F$ , $k$ , and $x$ ) in a given problem.
Use Hooke's Law to determine the force, displacement, or force constant for a given system.
Calculate the potential energy of a spring using Hooke's Law.

Review a worked example of using Hooke's Law to determine force constant in Example 16.1: How Stiff are Car Springs?.

We can also use Hooke's Law to determine the potential energy, or stored energy in a spring. Any deformed system, such as a pulled spring, has stored energy. We can calculate the potential energy of a spring using the following equation:

$\mathrm{PE_{el}}=\frac{1}{2}kx^2$

where $\mathrm{PE_{el}}$ is the elastic potential energy of the spring.

Review an example of calculating the potential energy of a spring in Example 16.2: Calculating Stored Energy: A Tranquilizer Gun Spring.

4i. Identify the fundamental physical properties of a simple pendulum, and describe the relationships among them

Describe and sketch a simple pendulum.
What forces are involved in the swinging of a simple pendulum?
How do you describe the period of a simple pendulum?

A simple pendulum consists of a small mass on the end of a string. The string can swing in an arc in the x-y axis. Figure 16.14 shows a figure of a simple pendulum swinging. With a simple pendulum, there are two main opposing forces. First, the weight of the small mass on the string acts as a force in the down direction. There is the restoring force from the string opposing the weight of the mass. As the pendulum swings, the restoring force relates to the displacement and the angle of the pendulum with respect to the x-y coordinate system:

$F=-mg\theta$ , where $m$ is mass, $g$ is gravity, and $\theta$ is angle

$\theta = \frac{s}{L}$

$F=\frac{-mg}{L}s$ ,

where $L$ is the length of the string on the pendulum.

Because $m$ , $g$ , and $L$ are constants, this takes the form of Hooke's Law: $F=-kx$ .

The period ( $T$ ) of a simple pendulum is the time it takes for the pendulum to swing. It is given by the following equation:

$T = 2\pi\sqrt{\frac{L}{g}}$

Review a worked example of using the pendulum equations to calculate the acceleration of a given pendulum's motion in Example 16.5: Measuring Acceleration Due to Gravity: The Period of a Pendulum.

Unit 4 Vocabulary

Acceleration
External force
Force constant
Free body diagram
Friction
Hooke's Law
Inertia
Kinetic friction
Law of Action and Reaction
Law of Inertia
Mass
Net force
Newton (N)
Newton's First Law of Motion
Newton's Second Law of Motion
Newton's Third Law of Motion
Normal force
Period
Potential energy
Restoring force
Simple pendulum
Static friction
System
Tension
Unit for force
Weight