Unit 7 practice test: Oscillations

At equilibrium the spring is at its relaxed length, so U_s = 0 and all the energy is kinetic. That is moment 3.

• Correct: Moment 3: Right: U_s = 0 means x = 0.
• Moment 1: At moment 1 K = 0: the block is at a turning point.
• Moment 2: At moment 2 the spring still holds 0.09 J, so the block is displaced.
• Moment 4: At moment 4 the spring holds 0.16 J.

U_s = ½kx² and the total is ½kA². At moment 2, U_s/E = 0.09/0.36 = 1/4, so x²/A² = 1/4 and x = A/2.

• Correct: A/2: Right: a quarter of the energy in the spring means half the amplitude.
• A/4: This treats U_s as proportional to x; it goes as x².
• 3A/4: This uses the kinetic share (3/4) as if it gave the position.
• A/√2: A/√2 is where the energy splits half and half.

E = ½kA². Doubling A multiplies E by 2² = 4: 4 × 0.36 J = 1.44 J.

• Correct: 1.44 J: Right: total energy goes as the amplitude squared.
• 0.72 J: This doubles E; it should be multiplied by 4 because E depends on A².
• 0.36 J: The total energy depends on the amplitude; it is not fixed by the spring.
• 0.18 J: A larger amplitude stores more energy, not less.

At Y the string pulls straight up and Earth pulls straight down, so no force acts along the arc. That is the equilibrium position: the bob would stay there if placed at rest.

• Correct: Y, where nothing pushes the bob along its path: Right: at the equilibrium position the restoring force along the path is zero.
• X, where the bob is momentarily at rest: Being at rest for an instant does not make a point the equilibrium position. At X a restoring force pulls the bob back toward Y.
• Z, where the bob stops before turning back: At Z the restoring force is largest, which is the opposite of equilibrium.
• Y, because the bob is moving fastest at that point: Y is the right point, but speed does not define equilibrium; the zero restoring force does.

Amplitude is the largest displacement from equilibrium, measured from Y. The bob reaches 8° on each side.

• Correct: 8°: Right: from Y to X (or to Z) is 8°.
• 16°: This is the angle from one extreme to the other, which is twice the amplitude.
• 4°: This halves the amplitude; the bob reaches the full 8° on each side.
• 0°: The amplitude is not the displacement at equilibrium; it is the largest displacement.

Along the arc, gravity has a part of size mg sin θ pointing back toward Y. For small angles sin θ ≈ θ (in radians), so the restoring force is close to mgθ = (mg/ℓ)s, proportional to the displacement s along the arc.

• Correct: At small angles the restoring force along the arc is close to proportional to the bob's displacement: Right: a restoring force proportional to the displacement is what SHM needs.
• The string pulls the bob toward the pivot with a force that stays the same at each point: The string force is not constant (it is largest at Y) and it points along the string, not along the path, so it is not the restoring force.
• The bob's weight pulls it down, and gravity is a force that is the same everywhere: A constant downward force alone does not give a force proportional to the displacement along the arc.
• The bob moves along a curved path, so it returns to its starting point each cycle: Returning to the start shows periodic motion, not simple harmonic motion.

60° is 1.05 rad but sin 60° = 0.87, so the restoring force (mg sin θ) is 17% smaller than a force proportional to θ would be. The bob still swings back and forth, but the motion is no longer a close match to SHM.

• Correct: The motion is periodic but noticeably less like SHM, because sin 60° is well below 60° in radians: Right: the small-angle approximation sin θ ≈ θ breaks down at large angles.
• The motion is still SHM, because a pendulum oscillates about Y at any angle: Oscillating about Y makes the motion periodic; SHM also needs the restoring force to be proportional to the displacement.
• The motion is no longer periodic, because the bob speeds up too much at Y: Without friction the bob repeats the same swing, so the motion is still periodic.
• The motion is closer to SHM, because a larger angle gives a larger restoring force: A larger force is not the test; the force must stay proportional to the displacement, and at 60° it falls short.