A Scotch yoke is a mechanism that transforms the circular motion of a crank into the reciprocating motion of a shaft (or vice versa). It has been used in a number of different internal combustion engines and in control valves. In the Scotch yoke shown, the acceleration of point A is defined by the relation a = -1.8 sinkt, where a and t are expressed in m/s² and seconds, respectively, and k = 3 rad/s. Knowing that x = 0 and v= 0.6 m/s when t= 0, determine the velocity and position of point A when t = 0.2 s.
A Scotch yoke is a mechanism that transforms the circular motion of a crank into the reciprocating motion of a shaft (or vice versa). It has been used in a number of different internal combustion engines and in control valves. In the Scotch yoke shown, the acceleration of point A is defined by the relation a = -1.8 sinkt, where a and t are expressed in m/s² and seconds, respectively, and k = 3 rad/s. Knowing that x = 0 and v= 0.6 m/s when t= 0, determine the velocity and position of point A when t = 0.2 s.
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
Section: Chapter Questions
Problem 1.1MA
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![### Scotch Yoke Mechanism
A Scotch yoke is a mechanism that converts the circular motion of a crank into the reciprocating motion of a shaft (or vice versa). It has been used in various internal combustion engines and control valves. Below is an illustration and description of a Scotch yoke mechanism along with a mathematical problem to help understand its motion.
#### Problem Description
In the Scotch yoke mechanism shown in the diagram, the acceleration of point \( A \) is defined by the relation:
\[ a = -1.8 \sin(kt) \]
where \( a \) and \( t \) are expressed in \( \text{m/s}^2 \) and seconds, respectively, and \( k = 3 \, \text{rad/s} \).
**Given:**
- \( x = 0 \)
- \( v = 0.6 \, \text{m/s} \) when \( t = 0 \)
**Find:**
1. The velocity of point \( A \).
2. The position of point \( A \) at \( t = 0.2 \, \text{s} \).
#### Diagram Explanation
The diagram features a mechanical setup with the Scotch yoke mechanism. Key parts labeled in the diagram are:
- **Point A**: The specific point on the yoke whose motion is analyzed.
- **Crank B**: Rotates to produce the reciprocating motion in the yoke.
- **Slider C**: A part of the yoke which slides due to the crank’s rotation.
- **Rod D**: Connects the crank B to the point A.
*[Note: No numerical labels were given in the problem for points B, C, and D.]*
#### Solution
**1. Velocity of point A**
Given the acceleration equation:
\[ a = -1.8 \sin(kt) \]
where \( k = 3 \, \text{rad/s} \).
To find the velocity at \( t = 0.2 \, \text{s} \), we need to integrate the acceleration with respect to time.
\[ a = \frac{dv}{dt} \]
\[ \frac{dv}{dt} = -1.8 \sin(3t) \]
Integrating both sides with respect to \( t \):
\[ v(t) = \int -1.8 \sin(3t) \,](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fce6d4303-574a-4287-a913-82cabc220c8a%2F4eeb232c-5fac-4a9b-ae31-5694262b2c81%2F0fsd55_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Scotch Yoke Mechanism
A Scotch yoke is a mechanism that converts the circular motion of a crank into the reciprocating motion of a shaft (or vice versa). It has been used in various internal combustion engines and control valves. Below is an illustration and description of a Scotch yoke mechanism along with a mathematical problem to help understand its motion.
#### Problem Description
In the Scotch yoke mechanism shown in the diagram, the acceleration of point \( A \) is defined by the relation:
\[ a = -1.8 \sin(kt) \]
where \( a \) and \( t \) are expressed in \( \text{m/s}^2 \) and seconds, respectively, and \( k = 3 \, \text{rad/s} \).
**Given:**
- \( x = 0 \)
- \( v = 0.6 \, \text{m/s} \) when \( t = 0 \)
**Find:**
1. The velocity of point \( A \).
2. The position of point \( A \) at \( t = 0.2 \, \text{s} \).
#### Diagram Explanation
The diagram features a mechanical setup with the Scotch yoke mechanism. Key parts labeled in the diagram are:
- **Point A**: The specific point on the yoke whose motion is analyzed.
- **Crank B**: Rotates to produce the reciprocating motion in the yoke.
- **Slider C**: A part of the yoke which slides due to the crank’s rotation.
- **Rod D**: Connects the crank B to the point A.
*[Note: No numerical labels were given in the problem for points B, C, and D.]*
#### Solution
**1. Velocity of point A**
Given the acceleration equation:
\[ a = -1.8 \sin(kt) \]
where \( k = 3 \, \text{rad/s} \).
To find the velocity at \( t = 0.2 \, \text{s} \), we need to integrate the acceleration with respect to time.
\[ a = \frac{dv}{dt} \]
\[ \frac{dv}{dt} = -1.8 \sin(3t) \]
Integrating both sides with respect to \( t \):
\[ v(t) = \int -1.8 \sin(3t) \,
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