The position of a mass oscillating on a spring is given by x = (5.6 cm) cos [2πt / (2.03 s)].
(a) What is the period of this motion?
(b) What is the first time the mass is at the position x = 0?
To find the period of the motion, we need to determine the value of time t that corresponds to one complete cycle of the cosine function. The period can be calculated using the formula T = 2π/ω, where T is the period and ω is the angular frequency.
(a) From the given equation x = (5.6 cm) cos [2πt / (2.03 s)], we can see that the angular frequency ω is equal to 2π / (2.03 s).
Using the formula T = 2π/ω, we can substitute the value of ω to find the period T:
T = 2π / (2π / (2.03 s))
T = 2.03 s
Therefore, the period of motion is 2.03 seconds.
(b) To find the first time the mass is at the position x = 0, we will set up the equation x = 0 and solve for t.
0 = (5.6 cm) cos [2πt / (2.03 s)]
Using the cosine function, we can solve for t:
cos [2πt / (2.03 s)] = 0
To find the first time t when the cosine function equals zero, we need to find the first zero-crossing of the cosine function.
cos θ = 0 when θ = (2n + 1)π/2, where n is an integer.
Therefore, we can set up the equation:
2πt / (2.03 s) = (2n + 1)π/2
Simplifying the equation:
t = (2n + 1)π / (2 * 2.03)
t = (2n + 1)π / 4.06
Since we are looking for the first time when x = 0, n = 0:
t = π / 4.06
Therefore, the first time the mass is at the position x = 0 is approximately t ≈ 0.772 seconds.
To find the answers to these questions, we need to understand the given equation and its variables.
The equation for the position of a mass oscillating on a spring is given by x = A cos(ωt + φ), where:
- x represents the position of the mass,
- A represents the amplitude of the oscillation,
- ω represents the angular frequency, and
- t represents time.
(a) To find the period of the motion, we can use the formula T = 2π/ω, where T represents the period and ω represents the angular frequency.
In the given equation x = (5.6 cm) cos [2πt / (2.03 s)], we can see that the angular frequency is ω = 2π/(2.03 s). We can substitute this value into the formula to find the period:
T = 2π / (2π / (2.03 s))
T = 2.03 s
Therefore, the period of this motion is 2.03 seconds.
(b) To find the first time the mass is at the position x = 0, we can set x equal to 0 and solve for t.
In the given equation x = (5.6 cm) cos [2πt / (2.03 s)], we can set x = 0:
0 = (5.6 cm) cos [2πt / (2.03 s)]
Now, we need to solve for t. To do this, we can use the inverse cosine function (cos^-1) to find the value of t when the cosine of that value is equal to zero.
cos^-1 (0) = 2πt / (2.03 s)
The inverse cosine of 0 is π/2, so we substitute this into the equation:
π/2 = 2πt / (2.03 s)
To isolate t, we can multiply both sides of the equation by (2.03 s) / (2π):
t = (2.03 s) / (2π) * (π/2)
t ≈ 0.645 seconds
Therefore, the first time the mass is at the position x = 0 is approximately 0.645 seconds.
(a) 2.03s
(b)1.015s