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Resultant Force

Consider a car that starts moving forward from rest. It starts with a speed of zero and continues to gain speed while the car experiences a force. This change in speed is the result of multiple forces acting on the car. All of these forces can be combined in such a way that they can all be represented by one…

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Resultant Force

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Jetzt kostenlos anmeldenConsider a car that starts moving forward from rest. It starts with a speed of zero and continues to gain speed while the car experiences a force. This change in speed is the result of multiple forces acting on the car. All of these forces can be combined in such a way that they can all be represented by one single force. This is the **resultant force**.

We call the single force that represents the effects of all of the other forces acting on an object, the **resultant force**.

Recall that a force is an interaction between objects that causes one or more of the objects to change their state of motion. It can simply be thought of as a push or pull. Imagine a large box on the floor; if you wanted to get the box to move, you would need to push or pull the box; that is, you need to apply a force. The Figure below shows a force$F$applied to a box that is on a horizontal, frictionless floor. The block moves in the direction of the force. Force is a vector, and can be represented by an arrow which indicates the size and direction of the force.

What would happen if another force were to be applied to the box in the opposite direction? If friction is introduced between the box and the floor, it would become more difficult to move the box across the floor. The next figure depicts this scenario, where an additional force, the friction force$f$, acts in the opposite direction of the force$F$.

The resultant force is now **less** than before because the two forces act in **opposite directions**. If two forces were acting in the **same direction** the resultant force would be **greater **than before.

The resultant force on an object is the vector sum of all of the forces acting on the object. This comes from the resultant force definition and can be illustrated with the following examples.

A cart is placed on a smooth, horizontal surface and is acted upon by two forces. The first force is$30\mathrm{N}$, acting to the right and the second force is$60\mathrm{N}$, also acting to the right. What is the resultant force acting on the cart?

The resultant force is the combined effect - the single force that results from the two applied forces acting on the cart. In this case, the two forces act in the same direction and are therefore added.

${F}_{res}=60\mathrm{N}+30\mathrm{N}\phantom{\rule{0ex}{0ex}}=90\mathrm{N}$

The figure below shows a free-body diagram of the cart with the two forces acting on it.

A cart is placed on a smooth, horizontal surface and is acted upon by two forces. The first force is$30\mathrm{N}$, acting to the right and the second force is$10\mathrm{N}$, acting to the left. What is the resultant force acting on the cart?

In this scenario, the two forces are acting in opposite directions and the resultant force is now the difference between the two forces. We, therefore, subtract one force from the other to obtain a resultant force.

${F}_{res}=30\mathrm{N}-10\mathrm{N}\phantom{\rule{0ex}{0ex}}=20\mathrm{N}\phantom{\rule{0ex}{0ex}}$

That is, the resultant force is$20\mathrm{N}$to the right. The effect felt by the cart is as if we replaced the two individual forces with a single force of magnitude$20\mathrm{N}$, acting to the right.

A police officer is standing stationary on level ground. The weight of the officer,$W$, is$800\mathrm{N}$. The reaction force that the ground exerts on the officer,$R$, is$800\mathrm{N}$. Calculate the resultant force exerted on the police officer.

In this case, we can subtract one force from the other to obtain a resultant force of 0 N to the right, as follows.

${F}_{res}=800\mathrm{N}-800\mathrm{N}\phantom{\rule{0ex}{0ex}}=0\mathrm{N}$

The weight of the police officer is balanced by the reaction force and the police officer neither moves up nor down. The resultant force on the officer is the difference between the forces (since they act in opposite directions) which is zero. When the resultant force on an object is$0\mathrm{N}$, the object will remain at its current velocity unless it is acted upon by an external force.

We can write a general equation for the resultant force${F}_{res}$on an object when multiple forces act on it in a **straight line**. Suppose we have two* *forces,${F}_{1}$and${F}_{2}$acting on an object in the **same direction**; the resultant force equation can be written as the **sum** of each of the forces as follows.

${F}_{res}={F}_{1}+{F}_{2}$

If two forces,${F}_{3}$and${F}_{4}$, act in a straight line but in opposite directions on an object, the resultant force equation is written as the difference between the two objects as below.

${F}_{res}={F}_{3}-{F}_{4}$

Let's look at an example of multiple forces on an object in different directions. The corresponding resultant force diagram can be seen in the figure below.

${F}_{1}$and${F}_{2}$act in the same direction and${F}_{3}$acts in the opposite direction to${F}_{1}$and${F}_{2}$. The resultant force equation is, therefore:

${F}_{res}={F}_{1}+{F}_{2}-{F}_{3}$

Recall that force is a vector quantity, so we use minus to indicate its direction, e.g. if we choose the right direction to be positive, any force to the left has a minus sign in front of its value. The magnitude of a resultant force ${F}_{res}$,* *is a scalar quantity, so in the case of multiple forces acting on an object, the magnitude of the resultant force is just the size of the force without its direction. That is, we drop the minus sign in front of the number (if there is one). An example of this is given below.

A cart is placed on a smooth, horizontal surface and is acted upon by two forces. The first force is a$40\text{N}$force to the right, and the second force is a$50\text{N}$force to the left. What is the magnitude of the resultant force acting on the cart? (Take the right direction to be positive.)

Taking the rightwards direction as positive, the calculation is as follows:

${F}_{res}=40\mathrm{N}-50\mathrm{N}\phantom{\rule{0ex}{0ex}}=-10\mathrm{N}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\mathrm{Magnitude}\mathrm{of}{F}_{res}=10\mathrm{N}$

The magnitude of the resultant force is the absolute value of$-10\text{N}$, which is simply$10\text{N}$. This means that the size of the resultant force is$10\text{N}$. Additionally, we can see that its direction is to the left.

From the example above, we can see that there is no resultant force when the forces on an object are balanced. Also, the direction of the resultant force on an object gives us the direction in which the object will move. All of the forces mentioned above are applied parallel to the floor or surface along which the object is able to move. What would happen if the force on an object were applied at some angle to the surface?

Imagine that a box is being pushed across a floor by a force$F$of magnitude$20\text{N}$, but the force is applied downward and to the right at an angle of 35° to the horizontal. We have to resolve the force into components, since the box moves horizontally along the surface.

We can draw a resultant force diagram to resolve$F$into a horizontal and vertical component as follows (not drawn to scale).

If the arrows in the diagram are drawn accurately and to scale, we can measure the vertical component of the force (by measuring the length of the vertical arrow) to be$11\text{N}$and the horizontal component (by measuring the length of the horizontal arrow) to be$16\text{N}$. We must make sure to draw them to scale. Our arrows should form a right-angle triangle, when the arrows representing the horizontal and vertical components of the resultant force are placed head to tail.

The vertical component of the force balances the normal contact force between the floor and the box. The horizontal component is responsible for moving the box horizontally across the floor. If we can resolve forces to find the components responsible for causing objects to move, we can find the resultant force for any number of forces acting on an object.

- The combined effect of many forces is known as the
**resultant force.** - When two forces act in the same direction on an object, the resultant force is the sum of the two forces.
- When two forces act in opposite directions on an object, the resultant force is the difference between the two forces.
- When the forces on an object are balanced, there is no resultant force on the object.
- For two
*F*_{1}and*F*_{2}, that act on an object in the same direction, the resultant force equation is ${F}_{res}={F}_{1}+{F}_{2}$. - For two
*F*_{3}and*F*_{4}, that act on an object in opposite directions, the resultant force equation is ${F}_{res}={F}_{3}-{F}_{4}$. - The magnitude of the resultant force is the value of the force without the minus sign. It is the size of the force.
- The direction of the resultant force on an object is the direction in which the object will move.
- We have to resolve the applied forces, when they are applied at different angles to the direction of motion.
- The component of the resultant force in the direction of motion is responsible for the motion.

The resultant force on an object is the vector sum of all the forces acting on that object.

The resultant force is the vector sum of the three forces acting on that object.

**F**_{R }= **F**_{1} + **F**_{2} + **F**_{3 }+ ... + **F**_{n}

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