PHYS161 - Midterm #2 Concept Review

|F_net|∆x

The work done on an object, where |F_net| is the magnitude of the net force and ∆x is the displacement of the object. Positive relationship with kinetic energy.

(1/2)mv²

Kinetic energy of an object, where m is the mass of the object and v is the velocity of the object. Positive relationship with work.

|F_net|∆x = ∆((1/2)mv²) or W_net = ∆K

Work-Kinetic Energy Theorem in 1-D, where the change in kinetic energy is equal to the work done on an object.

Do internal forces change the energy of the system?

No, because internal forces get cancelled out because fo Newton’s 3rd law, since internal forces are always in action-reaction pairs.

F_net · ∆r = ∆((1/2)mv²) = |F_net||∆r|cosθ

Work-Kinetic Energy Theorem in 2-D and 3-D, where the change in kinetic energy is equal to the dot product between the net force vector F_net and the displacement vector ∆r, which is also equal to the magnitude of the net force vector |F_net| times the magnitude of the displacement vector |∆r| times the cosine of the angle θ between the net force vector and the displacement vector.

Conservative Forces

Forces that are reversible and do an amount of work dependent only on the objects change in position, and not on how that specific change occurred, meaning that it is path independent. They are also forces which, when doing work on a system to change its kinetic energy, there is a description where that energy goes (potential energy). Some examples include gravity and spring forces.

Potential Energy

A function of position that can be stored, it reaches its maximum value if kinetic energy is equal to zero.

U_G = mgh

Gravitational Potential Energy, where m is the mass of the object, g is the gravitational constant (on Earth - 9.8 m/s² ), and h is the displacement in the y direction.

K_i + U_i = K_f + U_f = E_mech

Conservation of Mechanical Energy for a closed system with only conservative forces and the total work of nonconservative forces is zero.

U_S = (1/2)k(∆x)² = -W_S

Spring Potential Energy, where k is the spring coefficient (determines the stiffness of a spring) and ∆x is the displacement of the spring. Derived from the integral of the force vector F = −k∆x, which means that the work of the spring force is equal to - (1/2)k(∆x)². However, the work of a field is equal to negative potential energy, which is why the potential energy is positive.

-∆PE = W_C = ∆KE

Relationship between the change of potential energy, work of a conservative force, and the change in kinetic energy if only conservative forces do work. Based on Work-KE Theorem and Conservation of Mechanical Energy Theorem. Note that ∆U = ∆PE and ∆K = ∆KE.

K_i + U_i + W_nc = K_f + U_f

Work-Kinetic Energy Theorem, in which the work of nonconservative forces is not zero.

Nonconservative Forces

Forces that are path dependent and irreversible, in which energy is lost or dissipative in the form of heat or sound. Some examples include friction and air resistance.

W_f = f_k/s · ∆r = |f_k/s||∆r|cosθ = (μ_k/s(N))|∆r|cosθ

Work of a Friction Force, which is defined as a nonconservative force since heat is lost when friction is applied. A kinetic force has a magnitude of f_k = μ_k(N) where N is the normal force and μ_k is the coefficient of kinetic friction. A static force has a magnitude of f_s ≤ μ_s(N), where N is the normal force and μ_s is the coefficient of static friction.

External Force

A force that is applied onto an object or system from the outside, such as a push, a pull, a gravitational normal force, a contact force, or friction. Only net external forces can change an object’s overall motion, velocity, or momentum. They change an object’s total mechanical energy, which is why they are often nonconservative forces.

Internal Force

A force that only acts between the parts or components that make up that system, which means that they cancel out because of Newton’s 3rd Law due to the fact that all of these forces have action-reaction force pairs. Gravity and spring forces are both considered to be internal forces.

W_ext = ∆KE

The work done by external forces is equal to the change in kinetic energy, as potential energy does not exist.

r_cm = (Σ_im_ir_i)/(Σ_im_i) = (Σ_ir_i)/(N) = (∫︁rdm)/(∫︁dm)

Center of Mass, in which it can be found for individually for the x and y coordinates of position vector r. It is equal to the average location of the x and y coordinates divided by the sum of all the masses. If the masses are all identical, it becomes the average of the x and y coordinates, in which N is the number of masses. For a continous mass distribution, it averages the locations of infinitesimally small pieces of m, which becomes an integral.

a_cm = (F_net)/(M)

The acceleration of center mass, in which it is equal to the sum of all the external forces acting on the system divided by the total mass.

P = (dW)/(dt) = (∆W)/(∆t) = F · v =| F_net||∆r|cosθ

Power is the rate of change to work with respect to time, which implies that the integral of power is work from power in respect to time or that it is equal to the value of power times time (W_p = ∫︁Pdt = Pt). Power is also equal to a force vector dotted with a velocity vector, or the magnitude of the force vector |F| times the magnitude of the velocity vector |v| times the cosine of the angle θ between the force vector and the velocity vector. Power has the units of J/s or Watts (W). A broader definition of work is that it is the derivative of energy or the rate of change of energy over the time (P = (dE_system)/(dt) = (∆E_system)/(∆t)).

p = mv or F_net∆t = ∆(mv) = (dp)/(dt) = ∆p

Momentum of an object is equal to the mass of the object times the velocity of the object. Since velocity is a vector that points in a specific direction, momentum is also a vector that points in the same direction as the velocity vector. The total momentum of a system is found by taking the sum across each object’s momentum. Additionally, the net force on an object (or system of objects) times the time gives the change in momentum of that object (or system).

F_net∆t = ∫︂F_netdt = J

Impulse, or the net force on an object (or system of objects) times the time gives the change in momentum of that object (or system), is equal to power.

F_net∆t = ∆p = J

Impulse-Momentum Theorem, which tells us that the units for both impulse and momentum are kg * m/s. To find the impulse or fange in momentum, take the area under the curve of a force vs. time graph.

p_i,system = p_f,system = constant or m₁v₁ + m₂v₂ = m₁v'₁ + m₂v’₂

Conservation of Momentum, which states that momentum is conserved in the entire system if no net external forces are acting on the system. In collisions, the system is can be often treated as isolated during the short time interval of the collision, even if other steady forces are present.

Elastic Collision

A collision where total momentum is conserved and total kinetic energy is conserved. Objects bounce off of one another. Example: two rigid billiard balls

Inelastic Collision

A collision where total momentum is conserved but total kinetic energy is NOT conserved. Some kinetic energy is lost to heat, sound, deformation, etc. Example: two soft stress balls

Perfectly Inelastic Collision

A collision where total momentum is conserved where total kinetic energy is NOT conserved and instead, is minimized. Objects stick together and move with the same final velocity. Example: bullet embedding itself in a wooden block

When is conservation of energy useful?

Applies when there is no energy added or taken out of the system, most useful when work is easy to calculate.

When is conservation of momentum useful?

Only applies when there is no net force on the system. Usually useful for collisions and explosions. Useful for short periods of time (e.g. just before and after a collision) when there is no external force.

(1/2)m₁(v₁)² + (1/2)m₂(v₂)² = (1/2)m₁(v'₁)² + (1/2)m₂(v'₂)²

Kinetic Energy Conservation, it only occurs in elastic collisions.