Analyze Energy Using Conservation Laws - Physics
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Identify the correct condition for using $mgh$ for gravitational potential energy.
Identify the correct condition for using $mgh$ for gravitational potential energy.
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Near Earth with approximately constant $g$. $mgh$ assumes constant gravitational field strength.
Near Earth with approximately constant $g$. $mgh$ assumes constant gravitational field strength.
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What is the gravitational potential energy change when an object drops by height $h$?
What is the gravitational potential energy change when an object drops by height $h$?
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$\Delta U_g=-mgh$. Negative because potential energy decreases when falling.
$\Delta U_g=-mgh$. Negative because potential energy decreases when falling.
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What is the kinetic energy formula for a mass $m$ moving at speed $v$?
What is the kinetic energy formula for a mass $m$ moving at speed $v$?
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$K=\frac{1}{2}mv^2$. Energy of motion equals half the mass times velocity squared.
$K=\frac{1}{2}mv^2$. Energy of motion equals half the mass times velocity squared.
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What is the gravitational potential energy formula near Earth for mass $m$ at height $h$?
What is the gravitational potential energy formula near Earth for mass $m$ at height $h$?
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$U_g=mgh$. Weight ($mg$) times height gives gravitational potential energy.
$U_g=mgh$. Weight ($mg$) times height gives gravitational potential energy.
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What is the elastic potential energy formula for a spring with constant $k$ stretched by $x$?
What is the elastic potential energy formula for a spring with constant $k$ stretched by $x$?
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$U_s=\frac{1}{2}kx^2$. Spring energy equals half the spring constant times displacement squared.
$U_s=\frac{1}{2}kx^2$. Spring energy equals half the spring constant times displacement squared.
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What is the definition of total mechanical energy $E_{mech}$ in a system?
What is the definition of total mechanical energy $E_{mech}$ in a system?
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$E_{mech}=K+U$. Total mechanical energy is the sum of kinetic and potential energies.
$E_{mech}=K+U$. Total mechanical energy is the sum of kinetic and potential energies.
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What is the relationship between nonconservative work and mechanical energy change?
What is the relationship between nonconservative work and mechanical energy change?
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$W_{nc}=\Delta E_{mech}$. Nonconservative work equals the change in total mechanical energy.
$W_{nc}=\Delta E_{mech}$. Nonconservative work equals the change in total mechanical energy.
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What is the sign of work done by kinetic friction on a moving object over distance $d$?
What is the sign of work done by kinetic friction on a moving object over distance $d$?
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$W_f=-f_k d$. Friction opposes motion, so work is negative.
$W_f=-f_k d$. Friction opposes motion, so work is negative.
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What conservation equation applies when only conservative forces do work on a system?
What conservation equation applies when only conservative forces do work on a system?
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$K_i+U_i=K_f+U_f$. Mechanical energy is conserved when only conservative forces act.
$K_i+U_i=K_f+U_f$. Mechanical energy is conserved when only conservative forces act.
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What is the work–energy theorem relating net work and kinetic energy change?
What is the work–energy theorem relating net work and kinetic energy change?
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$W_{net}=\Delta K$. Net work equals the change in kinetic energy.
$W_{net}=\Delta K$. Net work equals the change in kinetic energy.
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What is the kinetic friction magnitude formula using coefficient $\mu_k$ and normal force $N$?
What is the kinetic friction magnitude formula using coefficient $\mu_k$ and normal force $N$?
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$f_k=\mu_k N$. Kinetic friction equals coefficient times normal force.
$f_k=\mu_k N$. Kinetic friction equals coefficient times normal force.
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What is the general energy accounting equation including nonconservative work?
What is the general energy accounting equation including nonconservative work?
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$K_i+U_i+W_{nc}=K_f+U_f$. Includes nonconservative work in energy conservation.
$K_i+U_i+W_{nc}=K_f+U_f$. Includes nonconservative work in energy conservation.
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Find the minimum initial speed $v_0$ needed to reach height $h$ if nonconservative work is $W_{nc}<0$.
Find the minimum initial speed $v_0$ needed to reach height $h$ if nonconservative work is $W_{nc}<0$.
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$v_0=\sqrt{2gh-\frac{2W_{nc}}{m}}$. Rearrange energy equation with negative nonconservative work.
$v_0=\sqrt{2gh-\frac{2W_{nc}}{m}}$. Rearrange energy equation with negative nonconservative work.
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Find the final speed $v_f$ if net work on a mass $m$ is $W_{net}$ and initial speed is $v_i$.
Find the final speed $v_f$ if net work on a mass $m$ is $W_{net}$ and initial speed is $v_i$.
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$v_f=\sqrt{v_i^2+\frac{2W_{net}}{m}}$. From work-energy theorem: $W_{net} = \frac{1}{2}m(v_f^2 - v_i^2)$.
$v_f=\sqrt{v_i^2+\frac{2W_{net}}{m}}$. From work-energy theorem: $W_{net} = \frac{1}{2}m(v_f^2 - v_i^2)$.
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Find the speed $v$ after dropping from rest through height $h$ with no losses.
Find the speed $v$ after dropping from rest through height $h$ with no losses.
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$v=\sqrt{2gh}$. From $mgh = \frac{1}{2}mv^2$ with energy conservation.
$v=\sqrt{2gh}$. From $mgh = \frac{1}{2}mv^2$ with energy conservation.
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Identify the correct expression for mechanical energy lost to friction over distance $d$ on level ground.
Identify the correct expression for mechanical energy lost to friction over distance $d$ on level ground.
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$\Delta E_{mech}=-\mu_k N d$. Friction force times distance gives energy lost.
$\Delta E_{mech}=-\mu_k N d$. Friction force times distance gives energy lost.
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Find the speed $v$ of a mass $m$ after descending height $h$ with kinetic friction doing work $-f_k d$.
Find the speed $v$ of a mass $m$ after descending height $h$ with kinetic friction doing work $-f_k d$.
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$v=\sqrt{2gh-\frac{2f_k d}{m}}$. Apply energy conservation with friction work included.
$v=\sqrt{2gh-\frac{2f_k d}{m}}$. Apply energy conservation with friction work included.
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Find the compression $x$ of a spring that stops a mass $m$ moving at speed $v$ on a frictionless surface.
Find the compression $x$ of a spring that stops a mass $m$ moving at speed $v$ on a frictionless surface.
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$x=v\sqrt{\frac{m}{k}}$. From $\frac{1}{2}mv^2 = \frac{1}{2}kx^2$ by energy conservation.
$x=v\sqrt{\frac{m}{k}}$. From $\frac{1}{2}mv^2 = \frac{1}{2}kx^2$ by energy conservation.
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Find the height $h$ reached if an object launched upward with speed $v_0$ stops at the top (no losses).
Find the height $h$ reached if an object launched upward with speed $v_0$ stops at the top (no losses).
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$h=\frac{v_0^2}{2g}$. From $\frac{1}{2}mv_0^2 = mgh$ at maximum height.
$h=\frac{v_0^2}{2g}$. From $\frac{1}{2}mv_0^2 = mgh$ at maximum height.
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Identify the correct energy equation for a pendulum bob between two heights when air resistance is negligible.
Identify the correct energy equation for a pendulum bob between two heights when air resistance is negligible.
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$mgh_i+\frac{1}{2}mv_i^2=mgh_f+\frac{1}{2}mv_f^2$. Conservation of mechanical energy for pendulum motion.
$mgh_i+\frac{1}{2}mv_i^2=mgh_f+\frac{1}{2}mv_f^2$. Conservation of mechanical energy for pendulum motion.
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Find the final speed $v$ if net work done on mass $m$ is $W_{\text{net}}$ and initial speed is $v_i$.
Find the final speed $v$ if net work done on mass $m$ is $W_{\text{net}}$ and initial speed is $v_i$.
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$v=\sqrt{v_i^2+\frac{2W_{\text{net}}}{m}}$. From work-energy theorem: $W_{\text{net}}=\frac{1}{2}m(v^2-v_i^2)$.
$v=\sqrt{v_i^2+\frac{2W_{\text{net}}}{m}}$. From work-energy theorem: $W_{\text{net}}=\frac{1}{2}m(v^2-v_i^2)$.
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Find the work done by friction $W_f$ if mechanical energy decreases from $E_i$ to $E_f$.
Find the work done by friction $W_f$ if mechanical energy decreases from $E_i$ to $E_f$.
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$W_f=E_f-E_i$. Friction work equals the mechanical energy loss.
$W_f=E_f-E_i$. Friction work equals the mechanical energy loss.
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Find the minimum initial speed $v_0$ to reach height $h$ if a constant friction force $f_k$ acts over distance $d$.
Find the minimum initial speed $v_0$ to reach height $h$ if a constant friction force $f_k$ acts over distance $d$.
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$v_0=\sqrt{2gh+\frac{2f_k d}{m}}$. Initial KE must overcome both PE gain and friction work.
$v_0=\sqrt{2gh+\frac{2f_k d}{m}}$. Initial KE must overcome both PE gain and friction work.
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Identify the sign of $\Delta U_g$ when an object moves downward by $\Delta h<0$.
Identify the sign of $\Delta U_g$ when an object moves downward by $\Delta h<0$.
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$\Delta U_g<0$. Moving down means $\Delta h<0$, so $mg\Delta h<0$.
$\Delta U_g<0$. Moving down means $\Delta h<0$, so $mg\Delta h<0$.
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What is the general energy accounting equation including nonconservative work $W_{\text{nc}}$?
What is the general energy accounting equation including nonconservative work $W_{\text{nc}}$?
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$K_i+U_i+W_{\text{nc}}=K_f+U_f$. Nonconservative work accounts for energy not conserved.
$K_i+U_i+W_{\text{nc}}=K_f+U_f$. Nonconservative work accounts for energy not conserved.
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What is the work done by kinetic friction with coefficient $\mu_k$ over distance $d$ on level ground?
What is the work done by kinetic friction with coefficient $\mu_k$ over distance $d$ on level ground?
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$W_f=-\mu_kmgd$. Friction opposes motion, so work is negative.
$W_f=-\mu_kmgd$. Friction opposes motion, so work is negative.
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What is the work done by a constant force $F$ over displacement $d$ at angle $\theta$?
What is the work done by a constant force $F$ over displacement $d$ at angle $\theta$?
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$W=Fd\cos\theta$. Work equals force times displacement times cosine of angle between them.
$W=Fd\cos\theta$. Work equals force times displacement times cosine of angle between them.
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State the work–energy theorem relating net work and kinetic energy change.
State the work–energy theorem relating net work and kinetic energy change.
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$W_{\text{net}}=\Delta K$. Net work equals the change in kinetic energy.
$W_{\text{net}}=\Delta K$. Net work equals the change in kinetic energy.
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State the conservation of mechanical energy equation when only conservative forces act.
State the conservation of mechanical energy equation when only conservative forces act.
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$K_i+U_i=K_f+U_f$. Total mechanical energy remains constant with only conservative forces.
$K_i+U_i=K_f+U_f$. Total mechanical energy remains constant with only conservative forces.
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What is the elastic potential energy stored in a spring with constant $k$ stretched by $x$?
What is the elastic potential energy stored in a spring with constant $k$ stretched by $x$?
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$U_s=\frac{1}{2}kx^2$. Spring PE equals half the spring constant times displacement squared.
$U_s=\frac{1}{2}kx^2$. Spring PE equals half the spring constant times displacement squared.
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