A force can be a function of position, velocity, or time. Most of the forces we consider in an introduction to dynamics are constant forces, but for this challenge question, imagine a block of mass m pushed horizontally on a frictionless surface for a period of time, τ, beginning with a force of zero, rising smoothly to a maximum, and then smoothly decreasing back to zero. What would be the speed of the object after the push was complete, and how far would it have travelled during the interaction? Starting with Newton’s Second Law, integrate each function (include in the picture both functions that need to be integrated) over this time to determine the final speed of the block after the force is no longer acting, assuming the block starts from rest. Then integrate (using the functions in the picture attached) each function again to determine the total distance travelled in each case. How close are the final calculated speeds and distances to each other? Let Fmax = 2.0 N, m = 0.5 kg, and τ = 2.00 seconds.
A force can be a function of position, velocity, or time. Most of the forces we consider in an introduction to dynamics are constant forces, but for this challenge question, imagine a block of mass m pushed horizontally on a frictionless surface for a period of time, τ, beginning with a force of zero, rising smoothly to a maximum, and then smoothly decreasing back to zero. What would be the speed of the object after the push was complete, and how far would it have travelled during the interaction? Starting with Newton’s Second Law, integrate each function (include in the picture both functions that need to be integrated) over this time to determine the final speed of the block after the force is no longer acting, assuming the block starts from rest. Then integrate (using the functions in the picture attached) each function again to determine the total distance travelled in each case. How close are the final calculated speeds and distances to each other? Let Fmax = 2.0 N, m = 0.5 kg, and τ = 2.00 seconds.
College Physics
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Chapter1: Units, Trigonometry. And Vectors
Section: Chapter Questions
Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...
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A force can be a function of position, velocity, or time. Most of the forces we consider in an introduction to dynamics are constant forces, but for this challenge question, imagine a block of mass m pushed horizontally on a frictionless surface for a period of time, τ, beginning with a force of zero, rising smoothly to a maximum, and then smoothly decreasing back to zero. What would be the speed of the object after the push was complete, and how far would it have travelled during the interaction?
Starting with Newton’s Second Law, integrate each function (include in the picture both functions that need to be integrated) over this time to determine the final speed of the block after the force is no longer acting, assuming the block starts from rest.
Then integrate (using the functions in the picture attached) each function again to determine the total distance travelled in each case.
How close are the final calculated speeds and distances to each other?
Let Fmax = 2.0 N, m = 0.5 kg, and τ = 2.00 seconds.
![First, let’s model the force function as the first half of a sine function:
\[
F(t) = F_{\text{max}} \sin \left(\frac{\pi t}{\tau}\right), \quad 0 < t > \tau
\]
For our second approximation, we can try an inverted parabola:
\[
F(t) = 4F_{\text{max}} \left[ \left(\frac{t}{\tau}\right) - \left(\frac{t}{\tau}\right)^2 \right], \quad 0 < t > \tau
\]
**Explanation:**
1. **Sine Function Model:**
- This equation models the force as a sine function, representing the first half of one sine wave. The force \( F(t) \) peaks at \( F_{\text{max}} \) and is defined for \( t \) between 0 and \( \tau \).
2. **Inverted Parabola Model:**
- This equation uses an inverted parabola to model the force function. The parabola is scaled by the factor \( 4F_{\text{max}} \) and it declines symmetrically after reaching a peak at the midpoint of interval \( \tau \).
These models can be used in various physics and engineering applications to approximate oscillating or time-dependent forces.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe799f64b-1f0b-4ff1-9ee3-5090ddf07729%2F219248fe-4226-4773-853f-cd3674ffb2fd%2F76cgloc_processed.png&w=3840&q=75)
Transcribed Image Text:First, let’s model the force function as the first half of a sine function:
\[
F(t) = F_{\text{max}} \sin \left(\frac{\pi t}{\tau}\right), \quad 0 < t > \tau
\]
For our second approximation, we can try an inverted parabola:
\[
F(t) = 4F_{\text{max}} \left[ \left(\frac{t}{\tau}\right) - \left(\frac{t}{\tau}\right)^2 \right], \quad 0 < t > \tau
\]
**Explanation:**
1. **Sine Function Model:**
- This equation models the force as a sine function, representing the first half of one sine wave. The force \( F(t) \) peaks at \( F_{\text{max}} \) and is defined for \( t \) between 0 and \( \tau \).
2. **Inverted Parabola Model:**
- This equation uses an inverted parabola to model the force function. The parabola is scaled by the factor \( 4F_{\text{max}} \) and it declines symmetrically after reaching a peak at the midpoint of interval \( \tau \).
These models can be used in various physics and engineering applications to approximate oscillating or time-dependent forces.
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