a weight on a vertical spring is given an initial downward velocity of 4cm/sec from a point 6 cm equilibrium. Assuming that the constant w has a value of 4, find the location of the weight 1 sec after it is set in motion.
To find the location of the weight 1 second after it is set in motion, we can use the equation for the displacement of a weight on a vertical spring:
x(t) = A * cos(wt + φ) + E
Where:
- x(t) is the displacement of the weight at time t
- A is the amplitude of the oscillation
- w is the angular frequency (w = 2πf)
- t is the time
- φ is the phase constant
- E is the equilibrium position of the weight
In this case, we are given:
- The constant w has a value of 4
- The weight has an initial downward velocity of 4 cm/sec
- The weight is initially 6 cm below the equilibrium position
From the given information, we can determine the values of A, φ, and E:
A = amplitude = maximum displacement = 6 cm
φ = phase constant = 0 (since the weight is released from equilibrium position)
E = equilibrium position = 0 (since the weight is initially 6 cm below the equilibrium position)
Now we can substitute these values into the equation to find the location of the weight 1 second after it is set in motion:
x(t) = 6 * cos(4t + 0) + 0
= 6 * cos(4t)
Therefore, the location of the weight 1 second after it is set in motion is given by:
x(1) = 6 * cos(4 * 1)
= 6 * cos(4)
≈ 6 * (-0.6536)
≈ -3.9216 cm
So, the weight is located approximately -3.9216 cm (below the equilibrium position) 1 second after it is set in motion.
To find the location of the weight 1 second after it is set in motion, we can use the equation for simple harmonic motion:
x(t) = A * cos(wt + φ)
Where:
- x(t) is the displacement of the weight from the equilibrium position at time t.
- A is the amplitude of the motion (maximum displacement).
- w is the angular frequency (related to the spring constant and mass).
- φ is the phase constant.
In this case, the weight starts with an initial downward velocity, which means that it is experiencing simple harmonic motion.
Given:
- The equilibrium position is 6 cm below the starting point.
- The constant w has a value of 4 (angular frequency).
To find the amplitude, we need to determine the maximum displacement of the weight. We know that the weight is 6 cm below the equilibrium position, so the amplitude is |6| = 6 cm.
Using the given constants, we can now write the equation for the motion:
x(t) = 6 * cos(4t + φ)
To find the phase constant φ, we need some additional information. This could be either the initial displacement or the initial velocity of the weight. Since the problem statement mentions the initial velocity, we can use that to determine φ.
Given:
- The weight has an initial downward velocity of 4 cm/sec.
At t = 0 (initial time), the velocity v(0) is given by:
v(t) = -Aw sin(wt + φ)
Substituting the given values:
4 = -6w sin(w * 0 + φ)
4 = -6w sin(φ)
From this equation, we can calculate φ by rearranging the equation and solving for sin(φ).
sin(φ) = -4 / (6w)
Using the given value of w (4), we have:
sin(φ) = -4 / (6 * 4)
sin(φ) = -1 / 6
Taking the inverse sine of -1 / 6, we find that φ ≈ -9.47 degrees.
Now we have determined both the amplitude (A = 6 cm) and the phase constant (φ ≈ -9.47 degrees). We can substitute these values back into the equation for x(t) to find the location of the weight 1 second after it is set in motion:
x(t) = 6 * cos(4t - 9.47)