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Magazine Chapter 13: Problem 30
A 0.84-kg air cart is attached to a spring and allowed to oscillate. If the displacement of the air cart from equilibrium is \(x=(10.0 \mathrm{cm}) \cos \left[\left(2.00 \mathrm{s}^{-1}\right) t+\pi\right],\) find (a) the maximum kinetic energy of the cart and (b) the maximum force exerted on it by the spring.
Short Answer
Expert verified
(a) KE_max = 0.0168 J (b) F_max = 0.336 N
Step by step solution
01
Identify the Given Values
We identify the given values from the problem. The mass of the air cart is \( m = 0.84 \, \text{kg} \). The displacement function is given as \( x(t) = (10.0 \, \text{cm}) \cos\left[(2.00 \, \text{s}^{-1}) t + \pi\right] \). Note: Convert displacement from cm to meters; \( x(t) = 0.10 \cos[(2.00) t + \pi] \).
02
Determine Maximum Velocity
The general form for displacement in simple harmonic motion is \( x(t) = A \cos(\omega t + \phi) \). Here, \( A = 0.10 \, \text{m} \), \( \omega = 2.00 \, \text{s}^{-1} \), and \( \phi = \pi \). The velocity is the first derivative of displacement with respect to time: \( v(t) = \frac{dx}{dt} = -A \omega \sin(\omega t + \phi) \). The maximum velocity \( v_{\text{max}} \) occurs when \( \sin(\omega t + \phi) = \pm 1 \). Thus, \( v_{\text{max}} = A \omega \). Calculate \( v_{\text{max}} = 0.10 \, \text{m} \times 2.00 \, \text{s}^{-1} = 0.20 \, \text{m/s} \).
03
Calculate Maximum Kinetic Energy
The formula for kinetic energy is \( KE = \frac{1}{2} mv^2 \). At maximum velocity, the kinetic energy is maximized: \( KE_{\text{max}} = \frac{1}{2} m v_{\text{max}}^2 \). Substitute \( m = 0.84 \, \text{kg} \) and \( v_{\text{max}} = 0.20 \, \text{m/s} \), to get \( KE_{\text{max}} = \frac{1}{2} \times 0.84 \, \text{kg} \times (0.20 \, \text{m/s})^2 = 0.0168 \, \text{J} \).
04
Determine Maximum Force by the Spring
In simple harmonic motion, the maximum force exerted by a spring is given by \( F_{\text{max}} = kA \), where \( k \) is the spring constant. The angular frequency \( \omega \) is related to the spring constant by \( \omega = \sqrt{\frac{k}{m}} \). Thus, the spring constant \( k \) can be calculated as \( k = m\omega^2 \). Substitute \( m = 0.84 \, \text{kg} \) and \( \omega = 2.00 \, \text{s}^{-1} \) to get \( k = 0.84 \, \text{kg} \times (2.00 \, \text{s}^{-1})^2 = 3.36 \, \text{N/m} \). Now, calculate \( F_{\text{max}} = k \times A = 3.36 \, \text{N/m} \times 0.10 \, \text{m} = 0.336 \, \text{N} \).
05
Conclusion
In summary, the maximum kinetic energy of the air cart is \( 0.0168 \, \text{J} \), and the maximum force exerted on it by the spring is \( 0.336 \, \text{N} \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Oscillation
In simple harmonic motion, oscillation refers to the repeated back-and-forth motion of an object about an equilibrium position. This type of motion is highly regular and predictable. The object moves due to restoring forces, like gravity or tension, that bring it back to its mean position. Oscillation can be seen in various scenarios like the swinging of pendulums or the bouncing of a spring.
- The amplitude of the oscillation is the maximum extent of the movement from the equilibrium.
- The time taken to complete one full oscillation cycle is known as the period.
- The frequency is the number of oscillations per unit time, typically measured in hertz (Hz).
- The shape of the oscillating motion in simple harmonic motion is often sinusoidal.
Kinetic Energy
Kinetic energy is the energy possessed by an object due to its motion. In the context of oscillation, the kinetic energy changes as the object moves. It is at its maximum when the object is moving at its fastest, which occurs as it passes through the equilibrium position.
The kinetic energy (\[KE\]) of an object can be calculated using the formula:\[KE = \frac{1}{2}mv^2\]where \(m\) is the mass of the object, and \(v\) is its velocity. For oscillating systems like springs, the velocity is highest at the equilibrium point, where the potential energy is lowest.
The kinetic energy (\[KE\]) of an object can be calculated using the formula:\[KE = \frac{1}{2}mv^2\]where \(m\) is the mass of the object, and \(v\) is its velocity. For oscillating systems like springs, the velocity is highest at the equilibrium point, where the potential energy is lowest.
- When kinetic energy is maximum, the speed of the object is highest.
- In simple harmonic motion, kinetic and potential energy continuously convert into each other.
- The total mechanical energy remains constant, reflecting the conservation of energy.
Spring Force
Spring force is the restoring force exerted by a spring. It is proportional to the displacement from the equilibrium position. This relationship is described by Hooke's Law, which states that the force required to extend or compress a spring by some distance \(x\) is proportional to that distance.
The formula for spring force (\[F\]) is:\[F = -kx\]where \(k\) is the spring constant, and \(x\) is the displacement from equilibrium. The negative sign indicates that the force exerted by the spring is in the opposite direction of the displacement.
The formula for spring force (\[F\]) is:\[F = -kx\]where \(k\) is the spring constant, and \(x\) is the displacement from equilibrium. The negative sign indicates that the force exerted by the spring is in the opposite direction of the displacement.
- The spring force is a central concept in simple harmonic motion, ensuring the periodic movement.
- A stiffer spring has a larger spring constant \(k\).
- Maximum spring force occurs at maximum displacement (amplitude).
Angular Frequency
Angular frequency is related to how often a cycle of oscillation occurs. It is often denoted by the Greek letter \(\omega\) and is measured in radians per second. Angular frequency gives a measure of the rate of oscillation, linking to terms like period and frequency.
The relationship is given by:\[\omega = 2\pi f\]where \(f\) is the frequency. Alternatively, the period \(T\), which is the time for one cycle, relates to angular frequency as:\[\omega = \frac{2\pi}{T}\]
The relationship is given by:\[\omega = 2\pi f\]where \(f\) is the frequency. Alternatively, the period \(T\), which is the time for one cycle, relates to angular frequency as:\[\omega = \frac{2\pi}{T}\]
- In harmonic systems like springs, knowing \(\omega\) helps predict motion.
- Increases in \(\omega\) result in more rapid oscillations.
- Angular frequency is crucial in defining other properties like kinetic energy and spring force.
Spring Constant
The spring constant is a measure of a spring's stiffness. Denoted by \(k\), it indicates how much force is needed to stretch or compress a spring by a unit length. This constant is a vital part of Hooke's Law, directly affecting the dynamics of oscillating systems.
The spring constant is linked with the angular frequency of an oscillating spring system via the formula:\[\omega = \sqrt{\frac{k}{m}}\]where \(m\) is the mass of the object attached to the spring. Larger values of \(k\) imply a stiffer spring, which influences both the angular frequency and the resulting oscillation.
The spring constant is linked with the angular frequency of an oscillating spring system via the formula:\[\omega = \sqrt{\frac{k}{m}}\]where \(m\) is the mass of the object attached to the spring. Larger values of \(k\) imply a stiffer spring, which influences both the angular frequency and the resulting oscillation.
- A high spring constant results in rapid oscillations.
- The spring constant is essential in calculating the maximum force exerted by the spring.
- A precise understanding of \(k\) allows predictions of system behavior under different conditions.
