The function y=Asin (sqrt(k/m)t) represents the oscillations of a mass m at the end of a spring. The constant k measures the stiffness of the spring.
1. Find the first positive time time t1 at which the mass is farthest from its equilibrium position: t1=_________________
2. Find the first positive time t2 at which the mass is moving fastest: t2=______________
3.Find the first positive time t3 at which the mass has the largest acceleration (in magnitude): t3=______________
4. What is the period, T , of the oscillation? T=________________
5. Find dT/dm: dT/dm=____________
For 1. I have (pi/2)/sqrt(k/m)
For 2 I have Acos((sqrt(k/m)))(sqrt(k/m))
For 3 I have Asqrt(k/m)
For 4 I don't have anything really...
For 5 m^(1/2)2pi/k^(1/2)
I don't know if these answers are right.. could someone please check these answers and help me with number 4?
Never mind, I got the answers
@Cady, what are the
1. To find the first positive time t1 at which the mass is farthest from its equilibrium position, we need to find the maximum value of the sine function. This occurs when the argument of the sine function is equal to π/2.
Asin(√(k/m)t) = π/2
Solving for t, we have:
√(k/m)t1 = π/2
t1 = (π/2) * √(m/k)
So the first positive time t1 at which the mass is farthest from its equilibrium position is t1 = (π/2) * √(m/k).
2. To find the first positive time t2 at which the mass is moving fastest, we need to find the maximum value of the cosine function. This occurs when the argument of the cosine function is equal to 0.
Acos(√(k/m)t2) = 0
Solving for t2, we have:
√(k/m)t2 = 0
t2 = 0
So the first positive time t2 at which the mass is moving fastest is t2 = 0.
3. To find the first positive time t3 at which the mass has the largest acceleration (in magnitude), we need to find the maximum value of the absolute value of the sine function. This occurs when the argument of the sine function is equal to π/2.
|Asin(√(k/m)t3)| = π/2
Since the absolute value of the sine function is always positive, we can simplify the equation to:
Asin(√(k/m)t3) = π/2
Solving for t3, we have:
√(k/m)t3 = π/2
t3 = (π/2) * √(m/k)
So the first positive time t3 at which the mass has the largest acceleration (in magnitude) is t3 = (π/2) * √(m/k).
4. The period T of the oscillation is the time it takes for one complete cycle. It is equal to the time it takes for the argument of the sine function to go from 0 to 2π (one full revolution). The argument of the sine function is given by √(k/m)t.
Therefore, we can set √(k/m)t = 2π and solve for t:
t = (2π) * √(m/k)
So the period T of the oscillation is T = (2π) * √(m/k).
5. To find dT/dm, we differentiate the equation for T with respect to m:
dT/dm = [(2π) * (√k) * (1/2) * (m^(-1/2))] / (2√(mk))
Simplifying, we have:
dT/dm = π / √(mkm)
So dT/dm = π / √(km^2)
Note: Please make sure to double-check these solutions as mistakes are possible.
To find the answers to the given questions, let's analyze the function y=Asin(sqrt(k/m)t) step by step:
1. Find the first positive time t1 at which the mass is farthest from its equilibrium position:
To find the time when the mass is farthest from its equilibrium position, we need to find the maximum value of the sine function. Since the maximum value of the sine function is 1, we have:
Asin(sqrt(k/m)t) = A
sin(sqrt(k/m)t) = 1
sqrt(k/m)t = pi/2 (since sin(pi/2) = 1)
t = (pi/2) * sqrt(m/k)
So the first positive time t1 at which the mass is farthest from its equilibrium position is t1 = (pi/2) * sqrt(m/k).
2. Find the first positive time t2 at which the mass is moving the fastest:
To find the time when the mass is moving the fastest, we need to find the maximum absolute value of the derivative of the function with respect to time (t). The derivative of y=Asin(sqrt(k/m)t) with respect to t is:
dy/dt = Asqrt(k/m)cos(sqrt(k/m)t)
To find the maximum absolute value of cos(sqrt(k/m)t), we know that the maximum value of the cosine function is 1, so:
|cos(sqrt(k/m)t)| = 1
Asqrt(k/m)cos(sqrt(k/m)t) = Asqrt(k/m)
Now, we need to find when exactly cos(sqrt(k/m)t) reaches its maximum absolute value of 1. This occurs when sqrt(k/m)t = 0, or t = 0. However, since we are looking for the first positive time, t2 will be the smallest positive value after t = 0. Therefore, we have t2 = 2pi/ sqrt(k/m).
3. Find the first positive time t3 at which the mass has the largest acceleration (in magnitude):
The acceleration is the second derivative of the function with respect to time (t). Taking the derivative of dy/dt = Asqrt(k/m)cos(sqrt(k/m)t) with respect to t, we get:
d^2y/dt^2 = -Asqrt(k/m)(sqrt(k/m))sin(sqrt(k/m)t) = -A(k/m)sin(sqrt(k/m)t)
To find the largest absolute value of sin(sqrt(k/m)t), we know that the maximum absolute value of sin function is 1, so:
|sin(sqrt(k/m)t)| = 1
-A(k/m)sin(sqrt(k/m)t) = A(k/m)
Now, we need to find when exactly sin(sqrt(k/m)t) reaches its maximum absolute value of 1. This occurs when sqrt(k/m)t = pi/2. Again, since we are looking for the first positive time, we have t3 = (pi/2) * sqrt(m/k).
4. The period T of the oscillation is the time it takes for one complete cycle. To find the period, we need to find the smallest positive value of t for which the function y = Asin(sqrt(k/m)t) repeats itself. The period can be obtained by dividing the full period 2pi by the frequency, which is sqrt(k/m). Therefore, we have T = (2pi) / sqrt(k/m) = 2pi * sqrt(m/k).
5. To find dT/dm, the derivative of the period T with respect to m, we can use the chain rule of differentiation. Recall that T = 2pi * sqrt(m/k). Taking the derivative of T with respect to m, we have:
dT/dm = d/dm (2pi * sqrt(m/k))
= (2pi/k) * (1/2) * (1/sqrt(m))
= pi / (k sqrt(m))
So, dT/dm = pi / (k sqrt(m)).
To summarize the answers:
1. The first positive time t1 at which the mass is farthest from its equilibrium position is t1 = (pi/2) * sqrt(m/k).
2. The first positive time t2 at which the mass is moving the fastest is t2 = 2pi / sqrt(k/m).
3. The first positive time t3 at which the mass has the largest acceleration (in magnitude) is t3 = (pi/2) * sqrt(m/k).
4. The period T of the oscillation is T = 2pi * sqrt(m/k).
5. The derivative of period T with respect to mass m is dT/dm = pi / (k sqrt(m)).