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Sliding movement with a motor. A heavy block of mass 1 kg is placed on a flat surface and attached to a horizontal spring. When the block is displaced 25 cm to the right of its equi- librium position the spring exerts a restoring force of 1 Newton to the left. Friction acts on the block proportional to its velocity such that when its speed is 1 m/s the block expe- riences a drag force of 4 Newtons. At time t = 0 seconds the block is at its equilibrium position traveling with a velocity of 1 m/s (i.e., traveling to the right). At t = 1 seconds a motor is switched on which exerts a force of t - 1 Newtons on the block. At t = 4 seconds the motor is switched off, and no external force acts on the block thereafter. (a) Rewrite the forcing function g(t) for the problem using unit step functions uc (t). Your answer should be expressible as a linear combination of uc(t)'s for different c's, each multiplied by some function of t. (b) Write down the initial value problem that models the horizontal position x(t) of the block, measured from its equilibrium position, for t≥ 0. (c) Find the Laplace transform X(s) of the solution of the initial-value problem from part (b). No need to find x(t).

Sliding movement with a motor. A heavy block of mass 1 kg is placed on a flat surface and attached to a horizontal spring. When the block is displaced 25 cm to the right of its equi- librium position the spring exerts a restoring force of 1 Newton to the left. Friction acts on the block proportional to its velocity such that when its speed is 1 m/s the block expe- riences a drag force of 4 Newtons. At time t = 0 seconds the block is at its equilibrium position traveling with a velocity of 1 m/s (i.e., traveling to the right). At t = 1 seconds a motor is switched on which exerts a force of t - 1 Newtons on the block. At t = 4 seconds the motor is switched off, and no external force acts on the block thereafter. (a) Rewrite the forcing function g(t) for the problem using unit step functions uc (t). Your answer should be expressible as a linear combination of uc(t)'s for different c's, each multiplied by some function of t. (b) Write down the initial value problem that models the horizontal position x(t) of the block, measured from its equilibrium position, for t≥ 0. (c) Find the Laplace transform X(s) of the solution of the initial-value problem from part (b). No need to find x(t).

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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by forcing function it means ay''+by'+cy=f(t)

a is mass

b is friction

c is constant of proportionality, or k

this is a math problem where you have to find an equation and it's initial conditions, and then find the laplace transform of the equation.

**Sliding Movement with a Motor** A heavy block of mass 1 kg is placed on a flat surface and attached to a horizontal spring. When the block is displaced 25 cm to the right of its equilibrium position, the spring exerts a restoring force of 1 Newton to the left. Friction acts on the block proportional to its velocity, such that when its speed is 1 m/s, the block experiences a drag force of 4 Newtons. - At time \( t = 0 \) seconds, the block is at its equilibrium position, traveling with a velocity of 1 m/s to the right. - At \( t = 1 \) second, a motor is switched on, exerting a force of \( t - 1 \) Newtons on the block. - At \( t = 4 \) seconds, the motor is switched off, and no external force acts on the block thereafter. **Tasks:** (a) **Forcing Function \( g(t) \)** - Rewrite the forcing function \( g(t) \) using unit step functions \( u_c(t) \) for different \( c's \), each multiplied by some function of \( t \). (b) **Initial Value Problem** - Write down the initial value problem that models the horizontal position \( x(t) \) of the block, measured from its equilibrium position, for \( t \geq 0 \). (c) **Laplace Transform \( X(s) \)** - Find the Laplace transform \( X(s) \) of the solution of the initial-value problem from part (b). There is no need to find \( x(t) \).
Transcribed Image Text:**Sliding Movement with a Motor** A heavy block of mass 1 kg is placed on a flat surface and attached to a horizontal spring. When the block is displaced 25 cm to the right of its equilibrium position, the spring exerts a restoring force of 1 Newton to the left. Friction acts on the block proportional to its velocity, such that when its speed is 1 m/s, the block experiences a drag force of 4 Newtons. - At time \( t = 0 \) seconds, the block is at its equilibrium position, traveling with a velocity of 1 m/s to the right. - At \( t = 1 \) second, a motor is switched on, exerting a force of \( t - 1 \) Newtons on the block. - At \( t = 4 \) seconds, the motor is switched off, and no external force acts on the block thereafter. **Tasks:** (a) **Forcing Function \( g(t) \)** - Rewrite the forcing function \( g(t) \) using unit step functions \( u_c(t) \) for different \( c's \), each multiplied by some function of \( t \). (b) **Initial Value Problem** - Write down the initial value problem that models the horizontal position \( x(t) \) of the block, measured from its equilibrium position, for \( t \geq 0 \). (c) **Laplace Transform \( X(s) \)** - Find the Laplace transform \( X(s) \) of the solution of the initial-value problem from part (b). There is no need to find \( x(t) \).
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