AP · AP C:M · free response
AP Physics C: Mechanics · Question 8
AP Physics C: Mechanics · Original GioPhysics question with a detailed, mark-by-mark answer guide.
- Demand
- discriminating
- Marks
- 12
- Topics
- 2
- Answer
- Complete
A particle of mass m moves along the x-axis in a conservative field described by the potential energy function U(x) = ax⁴ − bx², where a and b are positive constants.
- (a)
Derive Derive an expression for the force on the particle as a function of x.
2 marks - (b)
Determine Determine the positions of all equilibrium points, and state for each whether it is stable or unstable.
4 marks - (c)
Sketch Sketch a graph of U(x) against x, marking the equilibrium positions, and on the same axes indicate a total energy E for which the particle is confined to a region on one side of the origin only.
3 marks - (d)
Derive Derive an expression for the angular frequency of small oscillations of the particle about one of the stable equilibrium positions.
3 marks
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(a)
Derive Derive an expression for the force on the particle as a function of x.
List the given quantities with units, identify the required quantity, write the governing relationship before substituting, and keep extra digits until the final line so rounding does not distort the result.
- 1
1 point: uses F = −dU/dx
- 2
1 point: F = −4ax³ + 2bx
(b)
Determine Determine the positions of all equilibrium points, and state for each whether it is stable or unstable.
List the given quantities with units, identify the required quantity, write the governing relationship before substituting, and keep extra digits until the final line so rounding does not distort the result.
- 1
1 point: equilibrium requires F = 0, so 2bx = 4ax³
- 2
1 point: x = 0 and x = ±√(b/2a)
- 3
1 point: x = 0 is unstable — U has a local maximum there, since d²U/dx² = −2b < 0
- 4
1 point: x = ±√(b/2a) are stable, since d²U/dx² = 12ax² − 2b = 4b > 0 at those points
(c)
Sketch Sketch a graph of U(x) against x, marking the equilibrium positions, and on the same axes indicate a total energy E for which the particle is confined to a region on one side of the origin only.
List the given quantities with units, identify the required quantity, write the governing relationship before substituting, and keep extra digits until the final line so rounding does not distort the result.
- 1
1 point: symmetric double-well curve, with U → +∞ as x → ±∞ and a local maximum of U = 0 at x = 0
- 2
1 point: minima marked at x = ±√(b/2a), with U negative there
- 3
1 point: a horizontal line drawn at a negative energy E, between the minimum value of U and zero, with the two turning points on one side identified
(d)
Derive Derive an expression for the angular frequency of small oscillations of the particle about one of the stable equilibrium positions.
List the given quantities with units, identify the required quantity, write the governing relationship before substituting, and keep extra digits until the final line so rounding does not distort the result.
- 1
1 point: for small displacements about a minimum, the effective spring constant is k_eff = d²U/dx² evaluated at that point
- 2
1 point: k_eff = 12a(b/2a) − 2b = 4b
- 3
1 point: ω = √(k_eff/m) = 2√(b/m)
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