A simple harmonic oscillator consists of a spring with a force constant of k = 59.5 N/m connected to a block with a mass of m = 1.05 kg. Assume that the surface supporting the block is flat and frictionless, and there is no air resistance. At time t = 0, the spring is stretched 16.5 cm from its natural length and the block is moving at a speed of 1.95 m/s in the −x direction, as shown below. (a) Find x(t), the displacement of the block from equilibrium as a function of time. Hint: you’ll need to find the constants ω (in rad/s), A (in cm), ? (in radians) for the function: x(t) = Acos(ωt + φ). (b) What is the velocity of the block (in m/s) at t = 2.00 s? (c) What is the acceleration of the block (in m/s2) at t = 2.00 s?
Simple harmonic motion
Simple harmonic motion is a type of periodic motion in which an object undergoes oscillatory motion. The restoring force exerted by the object exhibiting SHM is proportional to the displacement from the equilibrium position. The force is directed towards the mean position. We see many examples of SHM around us, common ones are the motion of a pendulum, spring and vibration of strings in musical instruments, and so on.
Simple Pendulum
A simple pendulum comprises a heavy mass (called bob) attached to one end of the weightless and flexible string.
Oscillation
In Physics, oscillation means a repetitive motion that happens in a variation with respect to time. There is usually a central value, where the object would be at rest. Additionally, there are two or more positions between which the repetitive motion takes place. In mathematics, oscillations can also be described as vibrations. The most common examples of oscillation that is seen in daily lives include the alternating current (AC) or the motion of a moving pendulum.
A simple harmonic oscillator consists of a spring with a force constant of k = 59.5 N/m connected to a block with a mass of m = 1.05 kg. Assume that the surface supporting the block is flat and frictionless, and there is no air resistance. At time t = 0, the spring is stretched 16.5 cm from its natural length and the block is moving at a speed of 1.95 m/s in the −x direction, as shown below.
(a) Find x(t), the displacement of the block from equilibrium as a function of time. Hint: you’ll need to find the constants ω (in rad/s), A (in cm), ? (in radians) for the function: x(t) = Acos(ωt + φ).
(b) What is the velocity of the block (in m/s) at t = 2.00 s?
(c) What is the acceleration of the block (in m/s2) at t = 2.00 s?
![A simple harmonic oscillator consists of a spring with a force constant of \( k = 59.5 \, \text{N/m} \) connected to a block with a mass of \( m = 1.05 \, \text{kg} \). Assume that the surface supporting the block is flat and frictionless, and there is no air resistance. At time \( t = 0 \), the spring is stretched \( 16.5 \, \text{cm} \) from its natural length and the block is moving at a speed of \( 1.95 \, \text{m/s} \) in the \( -x \) direction, as shown below.
The diagram illustrates a block connected to a spring. The block is positioned on a horizontal line and is stretched to the right, indicating an initial displacement of \( 16.5 \, \text{cm} \) from equilibrium. The block is moving to the left at a speed of \( 1.95 \, \text{m/s} \).
(a) Find \( x(t) \), the displacement of the block from equilibrium as a function of time. Hint: you’ll need to find the constants \( \omega \) (in rad/s), \( A \) (in cm), \( \phi \) (in radians) for the function:
\[ x(t) = A \cos(\omega t + \phi). \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5b1907b1-a684-4a7a-a6d2-eae182fd8cd2%2F7f5b58fb-abd7-4811-8f2d-d98736b69b47%2F2gpv2il_processed.png&w=3840&q=75)
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