QUESTIONS (a) AND (b) ARE BASED ON THE INFORMATION AND DIAGRAM BELOW:
A 0.4-kilogram mass is oscillating on a spring that has a force constant of k = 1,000 newtons per meter.
Question(a)
Which of the following measurements would allow you to determine the maximum velocity experienced by the mass?
(A) No additional information is required.
(B) Minimum velocity
(C) Maximum acceleration
(D) None of these would allow you to determine maximum velocity.
▶️Answer/Explanation
Ans: (C) Maximum acceleration allows you to determine maximum displacement:
ma = kA
Knowing the amplitude allows you to determine easily the maximum speed via energy conservation:
\(\frac{1}{2}kA^{2}=\frac{1}{2}mv^{2}\)
Question(b)
Which of the following statements concerning the oscillatory motion described above is correct? (All statements refer to magnitudes.)
(A) The maximum velocity and maximum acceleration occur at the same time.
(B) The maximum velocity occurs when the acceleration is a minimum.
(C) The velocity is always directly proportional to the displacement.
(D) The maximum velocity occurs when the displacement is a maximum.
▶️Answer/Explanation
Ans:(B) Maximum velocities happen when going through the equilibrium point (zero acceleration and zero displacement): all KE and no PE.
Question
A pendulum of a given length swings back and forth a certain number of times per second. If the pendulum now swings back and forth the same number of times but in twice the time, the length of the pendulum should be
(A) doubled
(B) quartered
(C) quadrupled
(D) halved
▶️Answer/Explanation
Ans: (C) Doubling the period requires a quadrupling of length:
\(T=2\pi (\imath /g)^{1/2}\)
Question
Which of the following is the best method for finding a spring’s force constant \(k\) ?
(A) Hanging a known mass on the spring and dividing the weight by the length of spring
(B) Hanging several known masses on the spring, taking the average value of the mass, and dividing by the average length of the spring
(C) Hanging several known masses on the spring and finding the area under the curve after plotting force versus extension
(D) Hanging several known masses on the spring and finding the slope of the graph after plotting force versus extension
▶️Answer/Explanation
Ans:( D ) \(F=kx\). So when graphing \(f\) versus \(x\) , \(k\) will be the slope. The extension of the spring is \(x\) when \(F\) is the force applied to the spring.
Question
Which of the following is an equivalent expression for the maximum velocity attained by a mass \(m\) oscillating horizontally along a frictionless surface? The mass is attached to a spring with a force constant \(k\) and has an amplitude of \(A\) .
(A) \(Ak/m\)
(B) \(A(k/m)^{1/2}\)
(C) \(mg/kA\)
(D) \(A2k/m\)
▶️Answer/Explanation
Ans:(B) \(\frac{1}{2}kA^{2}=\) total energy when \(x = A\) (all PE, no KE)
\(\frac{1}{2}mv{_{max}}^{2}=\) total energy when \(x = 0\) (all PE, no KE)
Conservation of energy: \(\frac{1}{2}kA^{2}=\frac{1}{2}mv{_{max}}^{2}\)
\( v_{max}=(k/m)^{1/2}A\)
Question
A 0.5-kilogram mass is attached to a spring with a force constant of 50 newtons per meter. What is the total energy stored in the mass-spring system if the mass travels a distance of 8 cm in one cycle?
(A) 0.5 J
(B) 0.01 J
(C) 0.04 J
(D) 0.08 J
▶️Answer/Explanation
Ans:(B) In one cycle, the mass travels 4 amplitudes:
\(A\) = 0.02 m
Energy = \(\frac{1}{2}kA^{2}=\frac{1}{2}(50)(0.02)^{2}=0.01J\)