Spring Forces - AP Physics C: Mechanics
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What is the effect of negative sign in Hooke's Law?
What is the effect of negative sign in Hooke's Law?
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Indicates force direction is opposite to displacement. Ensures spring always opposes displacement from equilibrium.
Indicates force direction is opposite to displacement. Ensures spring always opposes displacement from equilibrium.
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What is the graphical representation of Hooke's Law?
What is the graphical representation of Hooke's Law?
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Linear graph of $F_s$ vs. $x$. Straight line with slope $-k$ passing through the origin.
Linear graph of $F_s$ vs. $x$. Straight line with slope $-k$ passing through the origin.
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What happens to the spring force if displacement is halved?
What happens to the spring force if displacement is halved?
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Spring force is also halved. Force is directly proportional to displacement per Hooke's Law.
Spring force is also halved. Force is directly proportional to displacement per Hooke's Law.
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If $k = 50 , \text{N/m}$ and $x = 0.3 , \text{m}$, find $F_s$.
If $k = 50 , \text{N/m}$ and $x = 0.3 , \text{m}$, find $F_s$.
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$F_s = -15 , \text{N}$. Calculate $F_s = -kx = -50 \times 0.3 = -15$ N.
$F_s = -15 , \text{N}$. Calculate $F_s = -kx = -50 \times 0.3 = -15$ N.
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How does the potential energy change if the spring constant doubles?
How does the potential energy change if the spring constant doubles?
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Potential energy also doubles. Since $U = \frac{1}{2}kx^2$, doubling $k$ doubles the energy stored.
Potential energy also doubles. Since $U = \frac{1}{2}kx^2$, doubling $k$ doubles the energy stored.
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Determine the spring constant if $F_s = -10 , \text{N}$ and $x = 0.25 , \text{m}$.
Determine the spring constant if $F_s = -10 , \text{N}$ and $x = 0.25 , \text{m}$.
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$k = 40 , \text{N/m}$. From $F_s = -kx$, solve $k = -\frac{F_s}{x} = -\frac{-10}{0.25} = 40$ N/m.
$k = 40 , \text{N/m}$. From $F_s = -kx$, solve $k = -\frac{F_s}{x} = -\frac{-10}{0.25} = 40$ N/m.
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Find $F_s$ if $k = 120 , \text{N/m}$ and $x = 0.05 , \text{m}$.
Find $F_s$ if $k = 120 , \text{N/m}$ and $x = 0.05 , \text{m}$.
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$F_s = -6 , \text{N}$. Apply $F_s = -kx = -120 \times 0.05 = -6$ N.
$F_s = -6 , \text{N}$. Apply $F_s = -kx = -120 \times 0.05 = -6$ N.
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Calculate the work done if $k=100 , \text{N/m}$ and $x$ changes from $0.1 , \text{m}$ to $0.2 , \text{m}$.
Calculate the work done if $k=100 , \text{N/m}$ and $x$ changes from $0.1 , \text{m}$ to $0.2 , \text{m}$.
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$W = -1.5 , \text{J}$. $W = -\Delta U = -(\frac{1}{2}k(0.2)^2 - \frac{1}{2}k(0.1)^2) = -1.5$ J.
$W = -1.5 , \text{J}$. $W = -\Delta U = -(\frac{1}{2}k(0.2)^2 - \frac{1}{2}k(0.1)^2) = -1.5$ J.
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Determine the spring constant if $F_s = -10 , \text{N}$ and $x = 0.25 , \text{m}$.
Determine the spring constant if $F_s = -10 , \text{N}$ and $x = 0.25 , \text{m}$.
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$k = 40 , \text{N/m}$. From $F_s = -kx$, solve $k = -\frac{F_s}{x} = -\frac{-10}{0.25} = 40$ N/m.
$k = 40 , \text{N/m}$. From $F_s = -kx$, solve $k = -\frac{F_s}{x} = -\frac{-10}{0.25} = 40$ N/m.
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What is the work done by a spring force?
What is the work done by a spring force?
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Negative change in potential energy, $-\Delta U$. Conservative force work equals negative change in potential energy.
Negative change in potential energy, $-\Delta U$. Conservative force work equals negative change in potential energy.
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Calculate the work done if $k=100 , \text{N/m}$ and $x$ changes from $0.1 , \text{m}$ to $0.2 , \text{m}$.
Calculate the work done if $k=100 , \text{N/m}$ and $x$ changes from $0.1 , \text{m}$ to $0.2 , \text{m}$.
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$W = -1.5 , \text{J}$. $W = -\Delta U = -(\frac{1}{2}k(0.2)^2 - \frac{1}{2}k(0.1)^2) = -1.5$ J.
$W = -1.5 , \text{J}$. $W = -\Delta U = -(\frac{1}{2}k(0.2)^2 - \frac{1}{2}k(0.1)^2) = -1.5$ J.
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What does it mean if a spring is 'ideal'?
What does it mean if a spring is 'ideal'?
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Obeys Hooke's Law perfectly. Force is exactly proportional to displacement with no deviations.
Obeys Hooke's Law perfectly. Force is exactly proportional to displacement with no deviations.
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What does the variable $k$ represent in Hooke's Law?
What does the variable $k$ represent in Hooke's Law?
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Spring constant. Measures stiffness of the spring in the force-displacement relationship.
Spring constant. Measures stiffness of the spring in the force-displacement relationship.
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What is the potential energy stored in a spring?
What is the potential energy stored in a spring?
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$U = \frac{1}{2}kx^2$. Energy stored when spring is compressed or stretched from equilibrium.
$U = \frac{1}{2}kx^2$. Energy stored when spring is compressed or stretched from equilibrium.
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State Hooke's Law formula for spring force.
State Hooke's Law formula for spring force.
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$F_s = -kx$. Force is proportional to displacement, with negative sign indicating restoring direction.
$F_s = -kx$. Force is proportional to displacement, with negative sign indicating restoring direction.
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Identify the unit of the spring constant $k$.
Identify the unit of the spring constant $k$.
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Newtons per meter (N/m). Force per unit displacement, combining force units (N) with length units (m).
Newtons per meter (N/m). Force per unit displacement, combining force units (N) with length units (m).
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Calculate the potential energy for $k=150 , \text{N/m}$ and $x=0.2 , \text{m}$.
Calculate the potential energy for $k=150 , \text{N/m}$ and $x=0.2 , \text{m}$.
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$U = 3 , \text{J}$. Use $U = \frac{1}{2}kx^2 = \frac{1}{2}(150)(0.2)^2 = 3$ J.
$U = 3 , \text{J}$. Use $U = \frac{1}{2}kx^2 = \frac{1}{2}(150)(0.2)^2 = 3$ J.
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What does the variable $x$ represent in Hooke's Law?
What does the variable $x$ represent in Hooke's Law?
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Displacement from equilibrium position. Distance from the natural length where spring force equals zero.
Displacement from equilibrium position. Distance from the natural length where spring force equals zero.
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What is the equilibrium position of a spring?
What is the equilibrium position of a spring?
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Position where net force is zero. Natural length where spring exerts no force on attached objects.
Position where net force is zero. Natural length where spring exerts no force on attached objects.
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What is the effect of mass on the period of a spring-mass system?
What is the effect of mass on the period of a spring-mass system?
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Period increases with mass. Heavier mass results in slower oscillation with longer period.
Period increases with mass. Heavier mass results in slower oscillation with longer period.
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If $k = 90 , \text{N/m}$ and $x = 0.15 , \text{m}$, find $U$.
If $k = 90 , \text{N/m}$ and $x = 0.15 , \text{m}$, find $U$.
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$U = 1.0125 , \text{J}$. Calculate $U = \frac{1}{2}kx^2 = \frac{1}{2}(90)(0.15)^2 = 1.0125$ J.
$U = 1.0125 , \text{J}$. Calculate $U = \frac{1}{2}kx^2 = \frac{1}{2}(90)(0.15)^2 = 1.0125$ J.
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Identify the phase of motion when a spring is at maximum extension.
Identify the phase of motion when a spring is at maximum extension.
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Maximum potential energy phase. All kinetic energy converted to potential energy at turning points.
Maximum potential energy phase. All kinetic energy converted to potential energy at turning points.
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What is the effect of increasing $k$ on the system's oscillation frequency?
What is the effect of increasing $k$ on the system's oscillation frequency?
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Oscillation frequency increases. Stiffer spring produces faster oscillations with higher frequency.
Oscillation frequency increases. Stiffer spring produces faster oscillations with higher frequency.
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Determine $k$ if $x = 0.4 , \text{m}$ and $U = 6 , \text{J}$.
Determine $k$ if $x = 0.4 , \text{m}$ and $U = 6 , \text{J}$.
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$k = 75 , \text{N/m}$. From $U = \frac{1}{2}kx^2$, solve $k = \frac{2U}{x^2} = \frac{2(6)}{(0.4)^2} = 75$ N/m.
$k = 75 , \text{N/m}$. From $U = \frac{1}{2}kx^2$, solve $k = \frac{2U}{x^2} = \frac{2(6)}{(0.4)^2} = 75$ N/m.
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What occurs when a spring is compressed?
What occurs when a spring is compressed?
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Potential energy increases. Compression stores elastic potential energy in the spring.
Potential energy increases. Compression stores elastic potential energy in the spring.
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Identify the unit of potential energy stored in a spring.
Identify the unit of potential energy stored in a spring.
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Joules (J). Energy unit, same as work and all other forms of energy.
Joules (J). Energy unit, same as work and all other forms of energy.
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What type of force is a spring force?
What type of force is a spring force?
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Restorative force. Always acts to restore the spring to its equilibrium position.
Restorative force. Always acts to restore the spring to its equilibrium position.
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Determine the force if $k = 200 , \text{N/m}$ and $x = 0.1 , \text{m}$.
Determine the force if $k = 200 , \text{N/m}$ and $x = 0.1 , \text{m}$.
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$F_s = -20 , \text{N}$. Apply $F_s = -kx = -200 \times 0.1 = -20$ N.
$F_s = -20 , \text{N}$. Apply $F_s = -kx = -200 \times 0.1 = -20$ N.
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Determine the potential energy if $x$ is zero.
Determine the potential energy if $x$ is zero.
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$U = 0 , \text{J}$. At equilibrium position, no energy is stored in the spring.
$U = 0 , \text{J}$. At equilibrium position, no energy is stored in the spring.
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Determine $x$ if $U = 2.5 , \text{J}$ and $k = 50 , \text{N/m}$.
Determine $x$ if $U = 2.5 , \text{J}$ and $k = 50 , \text{N/m}$.
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$x = 0.3162 , \text{m}$. Solve $x = \sqrt{\frac{2U}{k}} = \sqrt{\frac{2(2.5)}{50}} = 0.3162$ m.
$x = 0.3162 , \text{m}$. Solve $x = \sqrt{\frac{2U}{k}} = \sqrt{\frac{2(2.5)}{50}} = 0.3162$ m.
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