Why Isn't the Right Side Negative in a Vertical Spring-Mass System Equation?

  • #1
Ark236
26
3
Homework Statement
Hi everyone,

The problem has two parts. The first is to determine the equilibrium position of a mass attached to a spring. The second is to determine the equation of motion of the system, assuming that the block is pulled 1 cm down from its equilibrium position.
Relevant Equations
I choose the downward direction as positive. For the first part and using the FBD:

mg - k y_{0} = 0

Then the equilibrium position is y_{0} = mg/k.

For the second part, we have that:

mg -k y = m d^2 y/dt^2
I have a doubt with the last part. Why isn't the right side negative? Because when the block is released, it moves upwards.

thanks

image was obtained from here

Thanks.
C_3oscilador.png
 
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  • #2
Ark236 said:
Homework Statement: Hi everyone,

The problem has two parts. The first is to determine the equilibrium position of a mass attached to a spring. The second is to determine the equation of motion of the system, assuming that the block is pulled 1 cm down from its equilibrium position.
Relevant Equations: I choose the downward direction as positive. For the first part and using the FBD:

mg - k y_{0} = 0

Then the equilibrium position is y_{0} = mg/k.

For the second part, we have that:

mg -k y = m d^2 y/dt^2

I have a doubt with the last part. Why isn't the right side negative? Because when the block is released, it moves upwards.

thanks

image was obtained from here

Thanks.View attachment 334759
When ##y>y_0, mg-ky<0##, and then the right side, ##m \frac {d^2y} {dt^2}## is negative.
 
  • #3
Both sides are negative at that point. The net force points upwards and the body accelerates upwards. The equation shows that thenet force and the acceleration have the same sign. They are either both positive or both negative. If you put a minus sign in the equation itself, it would mean that the acceleration is in direction opposite to the net force. This would contradict Newton's second law, wouldn't?
 
  • #4
Ark236 said:
mg -k y = m d^2 y/dt^2

I have a doubt with the last part. Why isn't the right side negative? Because when the block is released, it moves upwards.
When the mass is below it's equilibrium position, which is bigger: mg or ky?
 

FAQ: Why Isn't the Right Side Negative in a Vertical Spring-Mass System Equation?

Why isn't the right side of the equation negative in a vertical spring-mass system?

In a vertical spring-mass system, the equation typically represents the forces acting on the mass. The right side of the equation often represents the restoring force of the spring, which is proportional to the displacement but acts in the opposite direction. If the displacement is taken as positive in the downward direction, the restoring force (upward) is negative. However, if the displacement is taken as positive in the upward direction, the restoring force (downward) is positive. The sign convention depends on the chosen coordinate system and direction of forces.

How does the coordinate system affect the sign in the spring-mass equation?

The coordinate system determines the sign convention for displacement and forces. In a vertical spring-mass system, if the upward direction is positive, the spring's restoring force (which acts upward when the mass is displaced downward) will be negative. Conversely, if the downward direction is positive, the restoring force will be positive when the spring is compressed or stretched downward. The equation's form depends on this choice of coordinate system.

What is the standard form of the vertical spring-mass system equation?

The standard form of the vertical spring-mass system equation is \( F = -kx \), where \( F \) is the restoring force, \( k \) is the spring constant, and \( x \) is the displacement from the equilibrium position. The negative sign indicates that the force acts in the opposite direction to the displacement. This form assumes that displacement is positive in the direction opposite to the restoring force.

Can the spring-mass system equation be written without a negative sign?

Yes, the spring-mass system equation can be written without a negative sign if the coordinate system is chosen such that the direction of displacement and the restoring force are aligned. For example, if downward displacement is positive and the restoring force is also considered positive in the downward direction, the equation would be \( F = kx \). The key is consistency in the chosen coordinate system and sign convention.

What role does gravity play in the vertical spring-mass system equation?

Gravity adds a constant force to the system, which shifts the equilibrium position of the mass. In the presence of gravity, the equilibrium position is where the spring force balances the gravitational force. The equation of motion for the mass includes both the spring force and the gravitational force. If \( y \) is the displacement from the equilibrium position, the equation can be written as \( F = -k(y - y_0) \), where \( y_0 \) is the displacement due to gravity. The sign convention must still be consistent with the chosen coordinate system.

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