Dynamics: Forces, Friction, Free Body Diagrams, and Tension

3.1 Laws of Motion

Newton's Laws of Motion

Newton's laws of motion are fundamental principles describing the relationship between the motion of an object and the forces acting upon it.

• Law of Inertia (Newton's First Law): A body at rest or moving at a constant speed will remain at rest or keep moving at a constant speed in a straight line unless a force is applied to it.

• Newton's Second Law: The acceleration of an object is directly proportional to the magnitude of net force applied.

3.2 Dynamics: Forces

Newton's Third Law of Motion

Newton's third law states that when two bodies interact, they exert forces on each other that are equal in magnitude and opposite in direction.

• Action and Reaction: For every action, there is an equal and opposite reaction.

• Examples:

  • Normal force

  • Weight

  • Applied force

Examples of Forces

• Example 1: If a block is pushed with a force of 50 N and continues to move constantly, the net force acting on the block is zero. There is no change in velocity.

• Example 2: If a block is pushed with 100 N of force, assuming gravity is 10 m/s2, the mass of the block is 10 kg. The force needed to move the block is 100 N.

Mass and Weight

• Mass: Measure of inertia; property of an object.

• Weight: Force exerted on an object by gravity; acts downward at the surface.

Normal Force

• Normal force: Perpendicular to the surface; the larger the weight, the larger the normal force.

3.2 Dynamics: Normal Force

Normal Force in Inclined Planes

The normal force is the component of contact force perpendicular to the surface.

• Example: A block on the floor with a mass of 20 kg is pushed by a force of 150 N in the horizontal direction. The normal force is equal to the weight of the block.

• Example: While the block is being pushed horizontally with a force of 150 N, the normal force remains equal to the weight of the block.

• Example: A block of 0.5 kg is resting on a horizontal surface. The normal force is equal to the weight.

3.2 Dynamics: Elevator Forces

Normal Force in Elevators

When a person stands in an elevator moving upward at constant speed, the normal force equals the force of gravity.

• Example: A person with a mass of 60 kg stands on an elevator. The normal force is equal to the weight: $F_N = mg = 60 \times 10 = 600\,N$

Normal Force on a Table

• Example: If a block is sitting on a table, the normal force is equal to the weight.

3.3 Free Body Diagrams

Free Body Diagrams

Free body diagrams are visual representations of all forces acting on an object. They are essential for analyzing the dynamics of a system.

• Key Forces:

  • Normal force ($F_N$)

  • Weight ($mg$)

  • Applied force ($F_{app}$)

  • Friction ($F_{fr}$)

Examples

• Example 1: A girl that weighs 500 N exerts a 300 N force on a box that weighs 1000 N. The normal force is equal to the weight of the box.

• Example 2: A boy is sliding down an inclined plane. The acceleration is determined by the net force along the incline.

3.4 Friction

Types of Friction

Friction is the force that opposes the motion of objects in contact. There are two main types:

• Static Friction: Frictional force between two surfaces that are not moving relative to each other. It increases until a maximum value is reached.

• Formula: $f_s \leq \mu_s F_N$

• Kinetic Friction: Force that opposes the motion of an object when it is already moving.

• Formula: $f_k = \mu_k F_N$

• Coefficient of Friction: $0 \leq \mu \leq 1$

Examples

• Example 1: A girl that weighs 500 N exerts a 300 N force on a box that weighs 1000 N. If the coefficient of static friction is 0.2, the box is in motion. The force of kinetic friction is $F_k = \mu_k F_N = 0.2 \times 1000 = 200\,N$

• Example 2: When someone is skating, kinetic friction acts on them.

• Example 3: A 20 kg box is sliding along a horizontal surface. The coefficient of kinetic friction is 0.2. The force of friction is $F_k = 0.2 \times 200 = 40\,N$

3.4 Tension

Tension in Strings and Cables

Tension is the force carried by a rope or cable attached to an object. It acts along the length of the rope and pulls equally on the objects at either end.

• Formula: $T = mg + ma$

• Direction: Tension is positive if it acts upward, negative if downward.

Examples

• Example 1: The acceleration of the system is $2\,m/s^2$ and the heavier mass moves up. If $m_1 = 4\,kg$ and $m_2 = 2\,kg$, the tension in the string attached to $m_2$ is $T_2 = m_2 g + m_2 a = 2 \times 10 + 2 \times 2 = 24\,N$

• Example 2: In a pulley system, 3 blocks are suspended in a line. The tension in the ropes from highest to lowest is $T_3 > T_2 > T_1$.

Additional info: Academic context and formulas have been expanded for clarity and completeness. All images included are directly relevant to the adjacent explanations.