2.7:
Two-Dimensional Force System: Problem Solving
Consider a wiper inclined at an angle of 25 degrees to the baseline 's' in a two-dimensional system. A force acting on the handle, pivoted to the center of the wiper, makes an angle of 20 degrees with the reference line 'c', which is parallel to baseline 's.'
If the force component along the a-axis is 65 Newtons, determine the magnitude of the force and its component along the b-axis.
Here, considering alternate angles, coa‘ becomes 25 degrees, while angle Foc is 20 degrees. So, the angle made by the force vector with the a-axis is determined by adding the two angles and is equal to 45 degrees.
As the force component along the a-axis is known, the magnitude of a force is estimated by the ratio of the a component of the force to the cosine of 45 degrees.
Now, the b component of the force is expressed as the product of the magnitude of the force and sine of 45 degrees. It is negative since it is along the negative b-axis.
2.7:
Two-Dimensional Force System: Problem Solving
Solving problems related to two-dimensional force systems is an essential aspect of mechanics and engineering. By applying the principles of vector analysis and force equilibrium, one can determine the effect of multiple forces acting on an object in a two-dimensional space.
The first step to solving a two-dimensional force system problem is to draw a free-body diagram of the object under consideration. This diagram helps identify all the external forces acting on the object, including their magnitudes, directions, and point of application.
Next, one needs to resolve the forces into their components in the x and y directions using the principles of vector analysis. This step helps convert the given forces into their Cartesian vector form, enabling easy representation and analysis. The components can be resolved using trigonometric functions, such as the sine and cosine of the given angles.
After breaking forces into components, the next step is determining the net force in each direction. This is done by adding up all of the forces in each direction. After that, the resultant force and its direction are determined using the Pythagorean theorem and trigonometry. If the net force is zero, the object is in equilibrium and not accelerating. If the net force is non-zero, then the object is accelerating in the direction of the net force. The principle of force equilibrium states that the sum of all external forces acting on the element must be equal to zero in both the x and y directions. This allows us to determine the unknown forces acting on the object, such as tension or compression in a member.
In some cases, the moment equilibrium principle is also used to determine the effect of external forces on the object. This principle states that the sum of all external moments acting on the element must equal zero. This step helps determine the values and directions of moments acting on the object.
Suggested Reading
- Beer, F.P.; Johnston, E.R.; Mazurek, D.F; Cromwell, P.J. and Self, B.P. (2019). Vector Mechanics for Engineers ‒ Statics and Dynamics. New York: McGraw-Hill. pp 29-33
- Hibbeler, R.C. (2016). Engineering Mechanics ‒ Statics and Dynamics. Hoboken, New Jersey: Pearson Prentice Hall. pp 33 ‒ 35.