Mass Passing Through Origin

[nycmathguy]

For part (d), the mass passes through the origin when t = 1, 3, 5, and 7. In other words, the odd numbers on the interval [0, 8].

When I replace t with 1, 3, 5, and 7, I get the point (0, 0). Can this be the reason for the textbook answer as stated by David Cohen?

 

[MathLover1]

no, you will not get the point (0, 0)

you will get:
s=4cos(pi*t/2)
t=1, 3, 5, and 7
s=4cos(pi*1/2)=0 -> point (t,s)=(1,0)
s=4cos(pi*3/2)=0 -> point (t,s)=(3,0)
s=4cos(pi*5/2)=0 -> point (t,s)=(5,0)
s=4cos(pi*7/2)=0 -> point (t,s)=(7,0)

points: (1,0) ,(3,0), (5,0),(7,0) -> s=0 for all as for the origin (0, 0)

 

[nycmathguy]

no, you will not get the point (0, 0)

you will get:
s=4cos(pi*t/2)
t=1, 3, 5, and 7
s=4cos(pi*1/2)=0 -> point (t,s)=(1,0)
s=4cos(pi*3/2)=0 -> point (t,s)=(3,0)
s=4cos(pi*5/2)=0 -> point (t,s)=(5,0)
s=4cos(pi*7/2)=0 -> point (t,s)=(7,0)

points: (1,0) ,(3,0), (5,0),(7,0) -> s=0 for all as for the origin (0, 0)

I don't get it. The point is clearly (t, s).
When t = odd numbers between 0 and 7, the value of s is 0. This I understand. Please, explain the textbook answer.

 

[MathLover1]

the mass is passing through the origin when s=0, which is t=1, t=3, t=5, and t=7

 

[nycmathguy]

the mass is passing through the origin when s=0, which is t=1, t=3, t=5, and t=7

Why is it important for us to know when the mass passes through the origin or moves right or left? Also, what is the MOST BASIC definition of mass? How common are problems involving mass found in a Calculus 3 textbook?

 

[MathLover1]

Periodic Motion-the usual physics terminology for motion

Newton’s first law implies that an object oscillating back and forth is experiencing forces.

Consider, for example, plucking a plastic ruler shown in the first figure. The deformation of the ruler creates a force in the opposite direction, known as a restoring force. Once released, the restoring force causes the ruler to move back toward its stable equilibrium position, where the net force on it is zero. However, by the time the ruler gets there, it gains momentum and continues to move to the right, producing the opposite deformation. It is then forced to the left, back through equilibrium, and the process is repeated until dissipative forces (e.g., friction) dampen the motion. These forces remove mechanical energy from the system, gradually reducing the motion until the ruler comes to rest.

Motion of a mass on an ideal spring: An object attached to a spring sliding on a frictionless surface is an uncomplicated simple harmonic oscillator. When displaced from equilibrium, the object performs simple harmonic motion that has an amplitude X and a period T. The object’s maximum speed occurs as it passes through equilibrium. The stiffer the spring is, the smaller the period T. The greater the mass of the object is, the greater the period T. (a) The mass has achieved its greatest displacement X to the right and now the restoring force to the left is at its maximum magnitude. (b) The restoring force has moved the mass back to its equilibrium point and is now equal to zero, but the leftward velocity is at its maximum. (c) The mass’s momentum has carried it to its maximum displacement to the right. The restoring force is now to the right, equal in magnitude and opposite in direction compared to (a). (d) The equilibrium point is reach again, this time with momentum to the right. (e) The cycle repeats.

problems involving mass found in a Calculus 3 are very common

 

[nycmathguy]

Periodic Motion-the usual physics terminology for motion

Newton’s first law implies that an object oscillating back and forth is experiencing forces.
View attachment 2157

Consider, for example, plucking a plastic ruler shown in the first figure. The deformation of the ruler creates a force in the opposite direction, known as a restoring force. Once released, the restoring force causes the ruler to move back toward its stable equilibrium position, where the net force on it is zero. However, by the time the ruler gets there, it gains momentum and continues to move to the right, producing the opposite deformation. It is then forced to the left, back through equilibrium, and the process is repeated until dissipative forces (e.g., friction) dampen the motion. These forces remove mechanical energy from the system, gradually reducing the motion until the ruler comes to rest.

Motion of a mass on an ideal spring: An object attached to a spring sliding on a frictionless surface is an uncomplicated simple harmonic oscillator. When displaced from equilibrium, the object performs simple harmonic motion that has an amplitude X and a period T. The object’s maximum speed occurs as it passes through equilibrium. The stiffer the spring is, the smaller the period T. The greater the mass of the object is, the greater the period T. (a) The mass has achieved its greatest displacement X to the right and now the restoring force to the left is at its maximum magnitude. (b) The restoring force has moved the mass back to its equilibrium point and is now equal to zero, but the leftward velocity is at its maximum. (c) The mass’s momentum has carried it to its maximum displacement to the right. The restoring force is now to the right, equal in magnitude and opposite in direction compared to (a). (d) The equilibrium point is reach again, this time with momentum to the right. (e) The cycle repeats.

problems involving mass found in a Calculus 3 are very common

Thank you for the study notes. Of course, Damped Harmonic Motion is more realistic. In DHM, an objects begins to move but eventually slows down and stops. In SHM, an object begins to move but never comes to a complete stop.