6.3 - Forces & Equilibrium
Free body diagrams
A free body diagram is a simplified drawing that shows an object as a single point or shape, with all the forces acting on it represented by arrows. These diagrams help visualize and analyze the forces affecting an object's motion or state of rest. By isolating the object and showing only the forces directly acting on it, free body diagrams make it easier to calculate net effects and determine if the object is in equilibrium.
Key elements of free body diagrams
- Object representation - Draw the object as a dot, box, or simple shape to focus attention on the forces rather than the object's details
- Force arrows - Each force is shown as an arrow starting from the object, with the arrowhead indicating the direction of the force
- Labeling - Clearly label each arrow with the type of force (such as weight or tension) and its magnitude if known
- Isolation - Only include forces acting directly on the object, ignoring forces the object exerts on other things
Free body diagrams are essential for breaking down complex situations into manageable parts, especially when multiple forces act simultaneously from different directions.
Representing forces with arrows
In free body diagrams, forces are represented using arrows where the length of the arrow indicates the magnitude (size) of the force, and the orientation of the arrow shows the direction in which the force acts. This visual representation allows for a quick understanding of how forces compare in strength and how they might balance or unbalance each other. Arrows are drawn to scale when possible, meaning a longer arrow represents a larger force.
How to draw force arrows:
- Start the arrow from the point where the force acts on the object.
- Point the arrowhead in the exact direction of the force's action.
- Adjust the arrow's length proportionally to the force's magnitude—for example, a force twice as large should have an arrow twice as long.
- Ensure arrows for opposing forces are drawn in opposite directions to show their competing effects.
This method helps in identifying whether forces are balanced or if there is a net effect in a particular direction.
Calculating resultant forces
The resultant force is the single net force that represents the combined effect of all individual forces acting on an object. To calculate it, separate the forces into horizontal and vertical components, then subtract opposing forces in each direction to find the net force. This process, called resolution of forces, simplifies analysis by treating horizontal and vertical directions independently.
Steps for calculating resultant forces:
- Identify all forces and draw a free body diagram with arrows.
- Resolve each force into its horizontal component (along the x-axis) and vertical component (along the y-axis) if they are not already aligned with these axes.
- For the horizontal direction: Add up all forces acting to the right and subtract those acting to the left to find the net horizontal force.
- For the vertical direction: Add up all forces acting upward and subtract those acting downward to find the net vertical force.
- Combine the net horizontal and vertical forces to get the overall resultant force, considering both magnitude and direction.
If forces are not at right angles, use trigonometry to find components, but focus on the subtraction of opposing forces as the key to determining the net effect.
Worked example - Calculating resultant force
An object experiences three forces: 10 N to the right, 6 N to the left, and 8 N upward. There are no other vertical forces. Calculate the resultant force by separating into horizontal and vertical components.
Step 1: Identify the components
- Horizontal forces: 10 N right, 6 N left
- Vertical forces: 8 N up, 0 N down
Step 2: Calculate net horizontal force
Net horizontal = 10 N - 6 N = 4 N to the right
Step 3: Calculate net vertical force
Net vertical = 8 N - 0 N = 8 N upward
Step 4: Determine overall resultant
The resultant force has a horizontal component of 4 N right and a vertical component of 8 N up. (Note: To find the exact magnitude and direction, further calculation like Pythagoras theorem could be used, but the components show the net effects.)
Conditions for equilibrium
Equilibrium occurs when the resultant force on an object is zero, meaning all forces acting on it are perfectly balanced. In this state, there is no net force in any direction, so the object remains at rest or continues moving at a constant velocity without acceleration. This balance requires that the sum of forces in the horizontal direction equals zero and the sum of forces in the vertical direction also equals zero.
Characteristics of equilibrium:
- Balanced forces - For every force in one direction, there is an equal and opposite force canceling it out
- Zero resultant - The net force is exactly zero, as calculated from components
- No change in motion - Objects in equilibrium do not speed up, slow down, or change direction
Equilibrium is a key concept in understanding stable systems, such as a book resting on a table where the weight downward is balanced by the normal force upward.
Non-zero resultant forces and acceleration
When the resultant force on an object is not zero, the forces are unbalanced, leading to acceleration according to Newton's second law. This law states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. As a result, a non-zero resultant force causes the object to change its velocity, either by speeding up, slowing down, or changing direction.
Newton's second law
Formula for Newton's second law:
Where:
- F = Resultant force (N)
- m = Mass of the object (kg)
- a = Acceleration (m/s2)
This equation shows how unbalanced forces produce motion changes, connecting the calculation of resultant forces to real-world effects like a pushed object starting to move.
Worked example - Calculating acceleration from resultant force
An object with a mass of 5 kg experiences a resultant force of 20 N to the right. Calculate the acceleration.
Step 1: Formula
Step 2: Rearrange for acceleration
Step 3: Substitution and calculation
The object accelerates at 4 m/s2 in the direction of the resultant force.