I need help with part b, c, and d please!
A 45 g mass is attached to a massless spring and allowed to oscillate around an equilibrium according to:
y(t) = 1.4 * sin( 4 * t ) where y is measured in meters and t in seconds.
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(a) What is the spring constant in N/m?
k = N/m *
0.72 OK
HELP: You are given m, the mass. What other quantity appears in the equation involving k, the spring constant, and m?
HELP: You are given the equation of motion
y(t) = A * sin( ù * t )
Now can you find the missing quantity?
HELP: Be careful of the units of m.
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(b) What is the Total Energy in the mass and spring in J?
E = J
HELP: The total energy is the sum of the kinetic energy and potential energy. At what point in the motion is the energy all kinetic? At what point is it all potential? Can you compute it at this point?
HELP: Kinetic energy is 1/2 * Mass * v2
Potential energy is 1/2 * k * y2
We know that at the mass's maximum y-position it has zero velocity, so simply compute Potential Energy at that point.
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(c) What is the maximum Kinetic Energy of the mass?
KE = J
HELP: Remember that Total Energy = Kinetic + Potential Energy.
HELP: When the mass is at y=0, it has zero potential energy. So what must its kinetic energy be at that point?
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(d) What is the maximum velocity of the mass in m/s ?
v(max) = m/s
HELP: Maximum velocity occurs at maximum kinetic energy.
HELP: KE = 1/2 * Mass * v2
Solve for v using the result from part (c).
for part b, take the amplitude (1.4) and square it and then multiply by what you got for part a.
so PE = .5 * k * x^2
= .5 * .72 * 1.4^2
for part c, the answer should be the exact same as part b.
and for part d, take what the answer you got for c and plug all your variables into the equation of
answer part c = .5 * .045 * v^2
To solve part (b), we need to find the total energy in the mass and spring system. The total energy is the sum of the kinetic energy and potential energy. At what point in the motion is the energy all kinetic? At what point is it all potential? Can you compute it at this point?
In this case, the maximum potential energy occurs when the mass is at its maximum displacement, which is the maximum value of y. At this point, the velocity is zero, so the kinetic energy is also zero. Therefore, the total energy is equal to the potential energy.
The potential energy of a mass-spring system is given by the formula:
Potential energy = 1/2 * k * y^2
Substituting the given equation for y(t) = 1.4 * sin(4 * t), we can find the maximum potential energy by plugging in the maximum value of y:
Potential energy = 1/2 * k * (1.4)^2
To solve part (c), we need to find the maximum kinetic energy of the mass. Remember that the total energy is equal to the sum of the kinetic and potential energy.
Thus, at the point when the mass is at y=0, it has zero potential energy. Therefore, the total energy is equal to the kinetic energy at this point. So the maximum kinetic energy is equal to the total energy we found in part (b).
To solve part (d), we need to find the maximum velocity of the mass. The maximum velocity occurs at the maximum kinetic energy.
The formula for kinetic energy is:
Kinetic energy = 1/2 * mass * velocity^2
Substituting the value of the maximum kinetic energy we found in part (c), we can solve for the maximum velocity:
Maximum velocity = √(2 * Maximum kinetic energy / mass)
By plugging in the values, you can calculate the maximum velocity in m/s.