solve this---Q1: The variation of acceleration, $a$ of a particle executing SHM with displacement $x$ is:
**Option 1 Graph Description:**
* Chart Type: 2D Cartesian graph.
* Coordinate Axes: Horizontal axis is $x$, labeled with an arrow pointing right. Vertical axis is $a$, labeled with an arrow pointing up. Axes intersect at the origin.
* Line: A curve starting from the origin (0,0) and extending into the first quadrant (positive $x$, positive $a$). The curve is concave down.
**Option 2 Graph Description:**
* Chart Type: 2D Cartesian graph.
* Coordinate Axes: Horizontal axis is $x$, labeled with an arrow pointing right. Vertical axis is $a$, labeled with an arrow pointing up. Axes intersect at the origin.
* Line: A straight line passing through the origin (0,0). The line extends into the second quadrant (negative $x$, positive $a$) and the fourth quadrant (positive $x$, negative $a$). The slope of the line is negative.
**Option 3 Graph Description:**
* Chart Type: 2D Cartesian graph.
* Coordinate Axes: Horizontal axis is $x$, labeled with an arrow pointing right. Vertical axis is $a$, labeled with an arrow pointing up. Axes intersect at the origin.
* Line: A curve starting from the origin (0,0) and extending into the first quadrant (positive $x$, positive $a$). The curve is concave up.
**Option 4 Graph Description:**
* Chart Type: 2D Cartesian graph.
* Coordinate Axes: Horizontal axis is $x$, labeled with an arrow pointing right. Vertical axis is $a$, labeled with an arrow pointing up. Axes intersect at the origin.
* Line: A straight line passing through the origin (0,0). The line extends into the first quadrant (positive $x$, positive $a$) and the third quadrant (negative $x$, negative $a$). The slope of the line is positive.
**Overall Layout:**
The options are presented in a 2x2 grid format, labeled numerically as 1, 2, 3, and 4. Option 1 is top-left, Option 2 is top-right, Option 3 is bottom-left, and Option 4 is bottom-right.
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Let's analyze the relationship between acceleration and displacement in Simple Harmonic Motion. The fundamental equation for SHM is a equals negative omega squared times x, where omega is the angular frequency. This equation tells us that acceleration is directly proportional to displacement but in the opposite direction.
Now let's examine the four graph options. Option 1 shows a concave down curve in the first quadrant. Option 2 shows a straight line with negative slope passing through the origin, extending into the second and fourth quadrants. Option 3 shows a concave up curve in the first quadrant. Option 4 shows a straight line with positive slope passing through the origin.
Let's analyze the SHM equation mathematically. The equation a equals negative omega squared times x is a linear equation of the form y equals mx, where y is a, x is x, and the slope m equals negative omega squared. Since omega squared is always positive, the slope is always negative. This means we have a straight line passing through the origin with a negative slope.
Now let's compare our analysis with the given options. Option 1 shows a non-linear curve, which doesn't match our linear equation. Option 3 also shows a non-linear curve. Option 4 shows a straight line but with a positive slope, which contradicts our negative slope requirement. Only Option 2 shows a straight line with negative slope passing through the origin, extending into the second and fourth quadrants. This perfectly matches our SHM equation. Therefore, the correct answer is Option 2.
To summarize what we have learned: Simple Harmonic Motion acceleration follows the equation a equals negative omega squared times x. This creates a linear relationship with negative slope. The graph passes through the origin and extends into the second and fourth quadrants. Option 2 correctly represents this fundamental relationship in physics.