Physics 211: Forces, Motion, Work, Energy, and Momentum – Comprehensive Study Notes

Forces and Motion

Newton's Laws of Motion

Newton's laws form the foundation for classical mechanics, describing the relationship between forces and the motion of objects.

• Newton's First Law (Law of Inertia): An object remains at rest or in uniform motion unless acted upon by a net external force.

• Newton's Second Law: The acceleration of an object is proportional to the net force acting on it and inversely proportional to its mass. $\vec{F}_{\text{net}} = m\vec{a}$

• Newton's Third Law: For every action, there is an equal and opposite reaction. $\vec{F}_{A \to B} = -\vec{F}_{B \to A}$

Example: A hanging weight attached to a wall and ceiling by ropes requires analyzing the tension in each rope using equilibrium conditions and force components.

Normal Force

The normal force is the perpendicular contact force exerted by a surface on an object in contact with it. It is a reaction force described by Newton's Third Law and acts at the point of contact.

Newton's Third Law in Everyday Life

Everyday activities such as walking or running involve pushing against the ground, which pushes back with an equal and opposite force, propelling us forward.

Horse and Cart Example

When a horse pulls a cart, the ground pushes the horse forward, while the horse pulls the cart. The forces between the horse and cart make them move together as a system.

Free-Body Diagrams

A free-body diagram isolates a single object and shows all external forces acting on it. Action-reaction pairs are not shown together since they act on different objects.

Momentum and Collisions

Conservation of Momentum

In the absence of external forces, the total momentum of a system remains constant. This principle is crucial in analyzing collisions and explosions.

• Momentum: $\vec{p} = m\vec{v}$

• Conservation Law: $\sum \vec{p}_{\text{initial}} = \sum \vec{p}_{\text{final}}$

Example: In the rifle problem, the rifle and bullet move in opposite directions after firing, sharing equal and opposite momentum.

Energy Sharing in Collisions

Although momentum is shared equally, kinetic energy is not. The lighter object (bullet) receives most of the kinetic energy due to its smaller mass.

Cart and Ball Example

When a person on a cart throws a ball, the cart moves in the opposite direction to conserve momentum. After the ball bounces back, the cart's motion changes again.

Elastic and Inelastic Collisions

• Elastic Collision: Both momentum and kinetic energy are conserved. For a light ball bouncing off a heavy wall, the ball reverses direction with nearly the same speed.

• Inelastic Collision: Objects stick together after collision, and some kinetic energy is transformed into other forms (e.g., heat, deformation).

Work and Kinetic Energy

Definition of Work

Work is done when a force causes displacement. Only the component of force in the direction of displacement contributes to work.

• $W = F \cos \phi \Delta x$

Example: Carrying a heavy object horizontally does no work against gravity since the force and displacement are perpendicular.

Work-Energy Theorem

The net work done on an object equals the change in its kinetic energy:

• $W_{\text{net}} = \Delta K = K_f - K_i$

Pulling a Box with and without Friction

When pulling a box, the work done by the applied force increases the box's kinetic energy. If friction is present, it does negative work, reducing the net energy gained by the box.

Stopping Distance from Energy

The stopping distance of a car can be calculated using the work done by friction to bring the car to rest:

• $\Delta x = \frac{v_0^2}{2\mu_k g}$

Friction and Inclined Planes

Static and Kinetic Friction

Friction opposes the relative motion of surfaces. The maximum static friction force determines when an object starts to slide:

• $f_{s,\text{max}} = \mu_s N$

On an inclined plane, the box starts sliding when the downslope component of gravity exceeds maximum static friction:

• $mg \sin \theta = \mu_s mg \cos \theta$

• $\tan \theta = \mu_s$

Mechanical Energy and Conservation

Potential and Kinetic Energy

Mechanical energy is the sum of kinetic and potential energy. In the absence of non-conservative forces (like friction), mechanical energy is conserved:

• $E = K + U = \text{constant}$

• Kinetic Energy: $K = \frac{1}{2}mv^2$

• Gravitational Potential Energy: $U = mgh$

Energy Conservation Applications

Energy conservation allows calculation of velocities and heights without considering the details of the motion path.

Roller Coaster Example

For a roller coaster starting at rest, the velocity at a lower point can be found using energy conservation:

• $v_f = \sqrt{2g\Delta y}$

Friction, Work, and Energy on an Incline

When friction is present, the work done by friction reduces the mechanical energy available for motion:

• $W = mg\sin\theta - \mu_k mg\cos\theta$

Potential Energy in Springs

Elastic Potential Energy

A spring stores potential energy when compressed or stretched. The work done by the spring is:

• $U = \frac{1}{2}kx^2$

Bow and Arrow Example

Drawing a bow stores elastic potential energy, which is converted to kinetic energy of the arrow upon release:

• $\frac{1}{2}kx^2 = \frac{1}{2}mv^2$

• $v = x\sqrt{\frac{k}{m}}$

Shooting Multiple Arrows

If multiple arrows are shot at once, the energy is divided among them, so each arrow receives less kinetic energy and travels a shorter distance.

Elastic Collision with a Spring

When a moving mass strikes a spring, its kinetic energy is converted into elastic potential energy at maximum compression, then back to kinetic energy as the spring recoils.

• $\frac{1}{2}mv_0^2 = \frac{1}{2}kx_f^2$

• $x_f = v_0\sqrt{\frac{m}{k}}$