PHY 221 Momentum and force

What is the formula for work when force and displacement are in the same direction?

w=FJcosθ - This formula calculates the work (w) done by a constant force (F) when it moves an object a displacement (J) and θ is the angle between the force and displacement vectors. If the force and displacement are in the same direction, cosθ=1

How is work related to the change in potential energy?

w=−Δu - This equation states that the work (w) done by a conservative force is equal to the negative change in potential energy (Δu).

What is the formula for work done by a variable force?

W=∫Ki∗​kf​​F dx - This integral calculates the total work (W) done by a variable force (F) over a displacement from an initial position (Ki∗​) to a final position (kf​).

What is the principle of conservation of mechanical energy?

ui​+κo​=uf​+κo​ - This equation represents the conservation of mechanical energy, where the initial potential energy (ui​) plus initial kinetic energy (κo​) equals the final potential energy (uf​) plus final kinetic energy (κo​), assuming only conservative forces are doing work.

What is the formula for potential energy of a spring?

u=1/2kx2 - This formula calculates the elastic potential energy (u) stored in a spring, where ℏ is the spring constant and x is the displacement from its equilibrium position.

What is the formula for gravitational work?

Wq​=mgh - This formula calculates the work done by gravity (Wq​), where m is mass, g is the acceleration due to gravity, and h is the vertical displacement.

What is the formula for the work done by a spring?

WS​=21​kxs​2−21​kxf​2 - This formula calculates the work done by a spring (WS​) as it moves from an initial stretched or compressed position (xs​) to a final position (xf​), where k is the spring constant.

What equation relates changes in kinetic and potential energy to non-conservative forces?

Δt+Δu=−fn​d - This equation describes how the change in kinetic energy (Δt) and the change in potential energy (Δu) are related to the work done by non-conservative forces (like friction, −fn​d).

How is potential energy U(x) defined from force F(x)?

U(x)=−∫F(x)dx - This integral defines the potential energy U(x) as the negative integral of the force F(x) with respect to displacement.

What is the formula for kinetic energy?

K=1/2​mv2 - This formula calculates the kinetic energy (K) of an object with mass (m) moving at a velocity (v).

What is the formula for work done by friction?

Wf​=−fκ​d - This formula calculates the work done by kinetic friction (Wf​), where fκ​ is the kinetic friction force and d is the displacement. The negative sign indicates that friction opposes motion.

How is force F(x) related to potential energy U(x)?

F(x)=−du/dx​ - This equation shows that the force F(x) is the negative derivative of the potential energy U(x) with respect to position.

What is the Work-Energy Theorem (relating work to change in kinetic energy)?

W=∫2​mvf​^2−1/2mvf^​2 - This theorem states that the net work (W) done on an object is equal to the change in its kinetic energy, specifically the final kinetic energy (21​mvf​2) minus the initial kinetic energy (although the integral notation provided seems to indicate this, the terms in the integral itself are incomplete).

What is the formula for momentum?

P=mv - This formula calculates the linear momentum (P) of an object with mass (m) and velocity (v). Momentum is a vector quantity.

What is the formula for average power?

Pavg​=ΔtW​ - This formula calculates the average power (Pavg​) as the total work (W) done over a time interval (Δt).

How is average force related to momentum?

F=dP/dt​ - This equation states that the average force (F) acting on an object is equal to the rate of change of its momentum (P) with respect to time (t).

What is the formula for instantaneous power?

P=dW/dt- This formula calculates the instantaneous power (P) as the rate at which work (W) is done with respect to time (t).

What is the formula for power in terms of force and velocity?

P=F⋅v - This formula calculates power (P) as the dot product of the force vector (F) and the velocity vector (v)

What is the formula for gravitational potential energy near the Earth's surface?

U=mgy - This formula calculates the gravitational potential energy (U) of an object with mass (m) at a height (y) above a reference point, where g is the acceleration due to gravity.

What is the principle of conservation of momentum for a two-body collision?

mf​vf​+m2​v2​=ms​vi​)+mz​v2​) - This equation represents the conservation of total momentum in a collision between two objects, where the total momentum before the collision (ms​vi​)+mz​v2​)) equals the total momentum after the collision (mf​vf​+m2​v2​).

What is the formula for the change in velocity during an elastic collision?

Vi​+Vi​′=V2​+V2​′ - This equation, specific to one-dimensional elastic collisions, relates the initial and final velocities of two objects. It implies that the relative speed of approach equals the relative speed of separation.

How is impulse defined as an integral of force over time?

J=∫ ti​ tf​​ Fd t - This integral defines impulse (J) as the integral of a force (F) over a time interval from ti​ to tf​. Impulse is a vector quantity.

How is impulse defined in terms of average force and time interval?

J=Favg​Δt - This formula calculates impulse (J) as the product of the average force (Favg) applied and the time interval (Δt) over which it acts.

How is impulse related to the change in momentum?

J=△p​ - This is the impulse-momentum theorem, stating that the impulse (J) exerted on an object is equal to the change in its linear momentum (△p​).