Defining Simple Harmonic Motion (SHM) AP Physics 1 MCQ – Exam Style Questions etc.
Defining Simple Harmonic Motion (SHM) AP Physics 1 MCQ
Unit 7: Oscillations
Weightage : 10-15%
Exam Style Practice Questions, Defining Simple Harmonic Motion (SHM) AP Physics 1 MCQ
Question
The figure above shows a pole with a spring around it and a \(2.5 \mathrm{~kg}\) block with a hole in the middle hanging from the spring. A light horizontal cord is attached to the block and a wall. The block is oscillating at \(10.0 \mathrm{~Hz}\), and the standing wave shown is formed.
The spring constant of the spring is approximately which of the following?
(A) \(10 \mathrm{~N} / \mathrm{m}\)
(B) \(100 \mathrm{~N} / \mathrm{m}\)
(C) \(1 \times 10^3 \mathrm{~N} / \mathrm{m}\)
(D) \(1 \times 10^4 \mathrm{~N} / \mathrm{m}\)
▶️Answer/Explanation
Ans:D
The angular frequency \( \omega \) of the oscillation is related to the frequency \( f \) by:
\[ \omega = 2 \pi f \]
For a mass-spring system, the angular frequency \( \omega \) is also given by:
\[ \omega = \sqrt{\frac{k}{m}} \]
where \( k \) is the spring constant.
Combining these equations:
\[ 2 \pi f = \sqrt{\frac{k}{m}} \]
\[ k = m (2 \pi f)^2 \]
\[ k = 2.5 \times (2 \pi \times 10.0)^2 \]
\[ k = 2.5 \times (20 \pi)^2 \]
\[ k \approx 2.5 \times 400 \times 9.87 \]
\[ k \approx 2.5 \times 4000 \]
\[ k \approx 10000 \text{ N/m} \]
Therefore, the spring constant \( k \) is approximately:(D)
Question
What additional measurement is needed to determine the speed of the wave on the cord?
(A) The thickness of the cord
(B) The unstretched length of the spring
(C) The average kinetic energy of the block
(D) The distance between nodes on the cord
▶️Answer/Explanation
Ans:D
The speed of a wave on a string is given by:
\[ v = \sqrt{\frac{T}{\mu}} \]
where \( T \) is the tension in the cord and \( \mu \) is the linear mass density of the cord.
To find the speed \( v \), we need to know the distance between the nodes, which will allow us to determine the wavelength \( \lambda \). For a standing wave, the wavelength \( \lambda \) and the frequency \( f \) can be used to find the speed \( v \) using the relationship:
\[ v = f \lambda \]
Thus, the additional measurement needed to determine the speed of the wave on the cord is: (D)
Question
A \(1.0 \mathrm{~kg}\) block is attached to an unstretched spring of spring constant \(50 \mathrm{~N} / \mathrm{m}\) and released from rest from the position shown in Figure 1 above. The block oscillates for a while and eventually stops moving \(0.20 \mathrm{~m}\) below its starting point, as shown in Figure 2. What is the change in potential energy of the block-spring-Earth system between Figure 1 and Figure 2 ?
(A) \(-1.0 \mathrm{~J}\)
(B) \(0 \mathrm{~J}\)
(C) \(1.0 \mathrm{~J}\)
(D) \(3.0 \mathrm{~J}\)
▶️Answer/Explanation
Ans:A
Question
A student is observing an object of unknown mass that is oscillating horizontally at the end of an ideal spring. The student measures the object’s period of oscillation with a stopwatch.
The student wishes to determine the spring constant of the spring using the measurements of the period of oscillation. Which of the following pieces of equipment would provide another measured quantity that is sufficient information to complete the determination of the spring constant?
(A) Meterstick
(B) Motion sensor
(C) Balance
(D) Photogate
▶️Answer/Explanation
Ans:C
To determine the spring constant \( k \) using the measurements of the period of oscillation, we need to use the formula for the period of a massspring system:
\[ T = 2\pi \sqrt{\frac{m}{k}} \]
Where:
\( T \) is the period of oscillation,
\( m \) is the mass of the oscillating object,
\( k \) is the spring constant.
To solve for the spring constant \( k \), we rearrange the formula:
\[ k = \frac{4\pi^2 m}{T^2} \]
From this equation, it is clear that in addition to the period \( T \), we need to know the mass \( m \) of the oscillating object to determine the spring constant \( k \).
Among the given options:
(A) Meterstick: This would help measure length, but it doesn’t directly help in determining the mass.
(B) Motion sensor: This could track the motion and help measure the period \( T \) accurately, but it doesn’t provide the mass.
(C) Balance: This is used to measure the mass of the object.
(D) Photogate: This could measure the timing and possibly the period \( T \), but it doesn’t provide the mass.
The correct piece of equipment that would provide the necessary information to determine the spring constant \( k \) is: (C) Balance
Question
A student is observing an object of unknown mass that is oscillating horizontally at the end of an ideal spring. The student measures the object’s period of oscillation with a stopwatch.
Using a number of measurements, the student determines the following.
The total energy of the object-spring system is most nearly
(A) \(0.98 \mathrm{~J}\)
(B) \(3.8 \mathrm{~J}\)
(C) \(7.6 \mathrm{~J}\)
(D) \(12.8 \mathrm{~J}\)
▶️Answer/Explanation
Ans:B
The total energy \( E \) of a mass-spring system is given by the sum of its potential energy at maximum displacement (amplitude \( A \)) and its kinetic energy at maximum speed \( v_{\text{max}} \). However, at maximum displacement, all the energy is potential, and at maximum speed, all the energy is kinetic. Thus, the total energy can be calculated using either the potential energy at the maximum displacement or the kinetic energy at the maximum speed.
The potential energy at maximum displacement is given by:
\[ E = \frac{1}{2} k A^2 \]
where:
\( k = 85 \, \mathrm{N/m} \) is the spring constant,
\( A = 0.30 \, \mathrm{m} \) is the amplitude.
Substitute the given values into the equation:
\[ E = \frac{1}{2} \times 85 \, \mathrm{N/m} \times (0.30 \, \mathrm{m})^2 \]
\[ E = \frac{1}{2} \times 85 \times 0.09 \]
\[ E = \frac{1}{2} \times 7.65 \]
\[ E = 3.825 \, \mathrm{J} \]