The position of a 20 g oscillating mass is given by x(t) = (2.0 cm) cos(2t), where t is in seconds. Determine the following.
The Amplitude (cm.)
The Period (s.)
The Spring Constant (N/m)
The maximum speed (cm/s)
The total energy (J)
The velocity at t=0.40 s. (cm/s)
I need your help, please help me guys. Thanks!
i figured out the amplitude and the period, i just don't kno how to get to the other 4 with using simple harmonic motion equations
can u give me these in numbers please, i don't understand what formulas u wrote and what some of the stuff in ur formulas are
how do yo find the spring constant?
Sure! I can help you with that. Let's go through each question step by step.
1. The Amplitude (cm):
The amplitude of an oscillating mass is the maximum displacement from its equilibrium position. In this case, the displacement is given as 2.0 cm, so the amplitude is also 2.0 cm.
2. The Period (s):
The period of an oscillating mass is the time it takes to complete one full cycle of motion. The period can be calculated using the formula T = 2π/ω, where ω is the angular frequency. In this case, the angular frequency is 2, so the period is T = 2π/2 = π seconds.
3. The Spring Constant (N/m):
The spring constant relates the displacement of the mass to the force exerted by the spring. In this case, the spring constant can be determined by using Hooke's Law, which states that F = -kx, where F is the force exerted by the spring, k is the spring constant, and x is the displacement. In this equation, we have F = mx", where m is the mass of the oscillating mass and x" is the second derivative of the displacement with respect to time. In this case, the mass is 20 g, or 0.02 kg, and the displacement equation is given as x(t) = (2.0 cm) cos(2t). We can take the second derivative of x(t) to find x":
x" = -4(2.0 cm) cos(2t)
Now we plug this into the equation F = mx":
mx" = -(0.02 kg) * 4(2.0 cm) cos(2t) = -0.16 cos(2t)
The force F is equal to -kx, so we solve for k:
-kx = -0.16 cos(2t)
Since this equation holds true for all values of t, we can equate the coefficients of cos(2t):
k = 0.16 N/cm
However, the SI unit of spring constant is N/m, so we need to convert cm to m:
k = 0.16 N/cm * (1 m / 100 cm) = 0.0016 N/m
Therefore, the spring constant is 0.0016 N/m.
4. The maximum speed (cm/s):
The maximum speed occurs at the maximum displacement, which is equal to the amplitude. We can find the maximum speed by taking the derivative of x(t) with respect to t:
v(t) = -2(2.0 cm) sin(2t)
Now we plug in t = 0 to find the maximum speed:
v(0) = -2(2.0 cm) sin(0) = 0 cm/s
Therefore, the maximum speed is 0 cm/s.
5. The total energy (J):
The total energy of the oscillating mass can be calculated using the equation E = (1/2)kA^2, where E is the total energy, k is the spring constant, and A is the amplitude. In this case, the spring constant is 0.0016 N/m and the amplitude is 2.0 cm, or 0.02 m. Plugging these values into the equation:
E = (1/2)(0.0016 N/m)(0.02 m)^2 = 3.2 x 10^-6 J
Therefore, the total energy is 3.2 x 10^-6 J.
6. The velocity at t = 0.40 s (cm/s):
To find the velocity at a specific time, we can take the derivative of x(t) with respect to t:
v(t) = -2(2.0 cm) sin(2t)
Plugging in t = 0.40 s:
v(0.40 s) = -2(2.0 cm) sin(2(0.40 s))
v(0.40 s) ≈ -1.165 cm/s
Therefore, the velocity at t = 0.40 s is approximately -1.165 cm/s.
I hope this helps! Let me know if you have any further questions.
spring constant:
Period=2PI sqrt (m/k)
max speed= 2*2
energy= 1/2 k x^2
velocity= 2*2 sin2t