Solve The Following Problems: The Criteria Used To Resolve M
Solve The Following Problems The Criteria Used To Resolve Must Be Sho
Solve the following problems, the criteria used to resolve must be shown. 1) A robot arm that controls the position of a video camera in a surveillance system is manipulated by a servomotor that exerts a force on the push bar. The force is given by: Where N and = .07m, x is the position of the end of the push rod. If the bar moves from x1=. 1m to x2=0.5 m How much work did the servomotor realized? Construct a graph of F vs. X, taking values x1 = 0.01 to x = 0.5.
Paper For Above instruction
The following analysis focuses on calculating the work done by a servomotor manipulating a robotic arm's push bar and constructing a force versus position graph. This problem encompasses fundamental principles of physics, specifically work and force relationships, applied within a robotics context. The core objective is to determine the work performed during the motion of the push bar from an initial position x₁ to a final position x₂, given a force function related to the position, and to visualize this relationship graphically.
Understanding the Force Function and Its Parameters
The problem states that the force exerted by the servomotor on the push bar is given by a particular formula involving parameters N and x, where N appears to be a scaling factor or constant, and x is the position of the end of the push rod. Although the exact form of the force function "F" is not explicitly provided in the prompt, contextually, it can be inferred that the force may depend linearly or non-linearly on x, often following a form similar to F(x) = N * x, or another functional relationship based on typical robotics and control system equations.
Given the mention of N with units in force (assuming Newtons, N), and the position x in meters, a common assumption is employing a linear force model, such as F(x) = N * x, where N is a constant representing force per unit length. Using the parameters in the problem, with N being a force constant and x the position, the force varies as the robot arm moves, affecting the work done by the servomotor.
Calculating the Work Done
The work done by a variable force in moving an object from position x₁ to x₂ is given by the integral of the force over the displacement:
Work (W) = ∫x₁x₂ F(x) dx
Assuming a linear force relationship F(x) = N * x, where N is 0.07 N/m (from the problem), then the integral becomes:
W = ∫x₁x₂ N * x dx
W = N * ∫x₁x₂ x dx
W = N * [x² / 2] evaluated from x₁ to x₂
Substituting the known values, x₁ = 0.1 m and x₂ = 0.5 m, and N = 0.07 N/m, we find:
W = 0.07 * ( (0.5)² / 2 - (0.1)² / 2 )
W = 0.07 * ( (0.25) / 2 - (0.01) / 2 )
W = 0.07 * (0.125 - 0.005)
W = 0.07 * 0.12
W = 0.0084 Joules
Thus, the servomotor performs approximately 0.0084 Joules of work moving the push bar from 0.1 m to 0.5 m.
Graphing F vs. x
To construct a graph of force (F) versus position (x), we plot F(x) = 0.07 x over the interval x = 0.01 m to x = 0.5 m. At x = 0.01 m, F = 0.07 0.01 = 0.0007 N. At x = 0.5 m, F = 0.07 * 0.5 = 0.035 N.
The graph is a straight line starting at (0.01, 0.0007) and ending at (0.5, 0.035), with a slope of 0.07. This illustrates how force increases linearly with the position of the push rod, consistent with the assumed linear model.
Discussion and Implications
The calculation demonstrates that the work done scales with the integral of force over displacement, highlighting the importance of understanding force profiles in robotic systems for efficiency and control optimization. The graphical representation assists in visualizing force variation and planning control strategies. If the force function were nonlinear, the integral would require different techniques, but the fundamental approach remains similar.
Conclusion
In conclusion, the servomotor performs approximately 0.0084 Joules of work during the specified movement, based on the assumed linear force model. The force vs. position graph depicts a direct proportionality, crucial for designing control algorithms and energy assessments in robotic systems. Precise modeling of force functions is essential for accurate work calculations and system optimization in robotics applications.
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