Physics quiz question: block stopped by compressing spring

Physics 205A Quiz 4, fall semester 2009
Cuesta College, San Luis Obispo, CA

Cf. Giambattista/Richardson/Richardson, Physics, 2/e, Problem 6.58, Comprehensive Problem 6.82

An 0.080 kg block traveling horizontally at 5.5 m/s hits a horizontal spring (k = 320 N/m) that is initially uncompressed. Neglect friction and drag. As a result the block is brought to a rest when the spring is compressed by:
(A) 0.037 m.
(B) 0.061 m.
(C) 0.087 m.
(D) 1.2 m.

Correct answer (highlight to unhide): (C)

Starting with the energy balance equation:

W_nc = ∆KE_tr + ∆PE_grav + ∆PE_elas,

where W_nc = 0 (no external gains/losses of mechanical energy), and ∆PE_grav = 0 (as there is no change in elevation of the block as it travels horizontally to come to rest), such that:

0 = ∆KE_tr + ∆PE_elas,

0 = (1/2)·m·∆(v²) + (1/2)·k·∆(x²),

0 = (1/2)·m·(v_f² – v₀²) + (1/2)·k·(x_f² – x₀²).

With initial parameters of v₀ = +5.5 m/s (block is moving) and x₀ = 0 (spring is relaxed/uncompressed), and final parameters v_f = 0 (block at rest), then the distance x_f that the spring has been compressed from equilibrium can now be solved for:

0 = (1/2)·m·(0² – v₀²) + (1/2)·k·(x_f² – 0²),

0 = –(1/2)·m·v₀² + (1/2)·k·x_f²,

k·x_f² = m·v₀²,

x_f = v₀·(m/k)^1/2,

x_f = (5.5 m/s)·((0.080 kg)/(320 N/m))^1/2 = 0.08696263565 m,

or to two significant figures, the spring is compressed by 0.087 m.

(Response (A) is (m·v₀/k)^1/2; response (B) is v₀·(m/(2·k))^1/2; response (D) is v₀·(2·g)^1/2.)

Sections 70854, 70855
Exam code: quiz04s7aT
(A) : 12 students
(B) : 16 students
(C) : 17 students
(D) : 3 students

Success level: 35%
Discrimination index (Aubrecht & Aubrecht, 1983): 0.37