Vertical Equilibrium & The Normal Force: Videos & Practice Problems

Equilibrium occurs when all forces acting on an object cancel out, resulting in zero acceleration. This means an object can still move at a constant velocity, as seen in examples with a box moving at 5 m/s and a book resting on a table. The normal force, which acts perpendicular to surfaces in contact, balances weight forces. Calculating the normal force involves using the equation $F=ma$, where acceleration is zero in equilibrium scenarios. Understanding these concepts is crucial for analyzing forces in various physical situations.

Equilibrium

Understanding equilibrium is crucial in physics, particularly when analyzing forces acting on an object. An object is said to be in equilibrium when all the forces acting on it cancel each other out, resulting in a net force of zero. This can be expressed mathematically using Newton's second law, \( F = ma \), where the sum of all forces (\( \Sigma F \)) equals zero, leading to an acceleration (\( a \)) of zero as well.

For example, consider a box being pulled by two equal forces in opposite directions. If each force is 10 Newtons, the net force is calculated as follows:

\[ \Sigma F = 10 \, \text{N} - 10 \, \text{N} = 0 \, \text{N} \]

Since the net force is zero, the acceleration of the box is also zero, confirming that it is in equilibrium. Importantly, equilibrium does not imply that the object is stationary; it can be moving at a constant velocity, such as 5 meters per second, without accelerating.

In another scenario, consider a 2-kilogram book resting on a table. The weight of the book can be calculated using the formula:

\[ W = mg \]

where \( m \) is the mass (2 kg) and \( g \) is the acceleration due to gravity (approximately 9.8 m/s²). Thus, the weight force is:

\[ W = 2 \, \text{kg} \times 9.8 \, \text{m/s}^2 = 19.6 \, \text{N} \]

In this case, the normal force exerted by the table also equals 19.6 N, acting in the opposite direction. The forces acting on the book can be summarized as:

\[ \Sigma F = N - mg = 0 \]

Since the book is at rest, the acceleration is zero, confirming that the forces are balanced and the object is in equilibrium.

In summary, equilibrium can be approached from two perspectives: knowing that the forces cancel implies zero acceleration, and knowing that acceleration is zero implies that the forces must cancel. This duality is essential for solving various physics problems related to forces and motion.

A 3-kg box of junk is being lowered on a string at a constant speed. What is the tension in the string?

• A. 0 N
• B. 3 N
• C. 29 N
• D. 32 N

Answer: (C)

Multiple Cables on a Loudspeaker

In this problem, we analyze a loudspeaker suspended by four vertical cables, each exerting a tension of 30 newtons. To determine the mass of the loudspeaker, we start by constructing a free body diagram, which helps visualize the forces acting on the system. The weight of the loudspeaker, represented as \( mg \) (where \( m \) is the mass and \( g \) is the acceleration due to gravity, approximately \( 9.8 \, \text{m/s}^2 \)), acts downward, while the tensions from the cables act upward.

Since there are four cables, the total upward force can be expressed as \( 4T \), where \( T \) is the tension in each cable. Given that \( T = 30 \, \text{N} \), the total upward force becomes \( 4 \times 30 = 120 \, \text{N} \). In equilibrium, the sum of the forces must equal zero, leading to the equation:

\[ 4T - mg = 0 \]

Rearranging this gives:

\[ 4T = mg \]

To find the mass \( m \), we can isolate it:

\[ m = \frac{4T}{g} \]

Substituting the known values:

\[ m = \frac{4 \times 30}{9.8} \approx 12.12 \, \text{kg} \]

This calculation shows that the mass of the loudspeaker is approximately 12.12 kilograms. The reasoning behind the distribution of tension among the cables is that as more cables are used, the weight supported by each cable decreases, allowing for a stable equilibrium without any acceleration, confirming that the system is at rest.

The Normal Force

The normal force is a fundamental concept in physics, particularly in mechanics, representing the force exerted by a surface to support the weight of an object resting on it. This force acts perpendicular to the surface, hence the term "normal," which is derived from the mathematical definition of perpendicularity (90 degrees). The normal force is typically denoted by the symbol N or sometimes f_N.

When analyzing forces acting on an object, it is essential to recognize that the normal force counteracts the weight of the object, which is calculated using the equation w = mg, where w is the weight, m is the mass, and g is the acceleration due to gravity (approximately 9.8 m/s²). For example, a book with a mass of 2.04 kg has a weight of w = 2.04 \times 9.8 = 20 \, \text{N}.

In scenarios where the object is at rest on a horizontal surface, the normal force equals the weight of the object, leading to the equation N = mg. However, when additional forces are applied, the normal force must be recalculated. For instance, if a downward force is applied to the book, the normal force increases to balance both the weight and the applied force. This can be expressed as N = mg + F, where F is the applied force. Conversely, if an upward force is applied that is less than the weight, the normal force will be less than the weight, calculated as N = mg - F.

In cases where the applied force exceeds the weight of the object, the normal force becomes zero, indicating that the object is no longer in contact with the surface. This situation can be analyzed using Newton's second law, F = ma, where the net force acting on the object determines its acceleration. For example, if a force of 30 N is applied upwards to the book, the resulting acceleration can be calculated as follows:

F_net = F - mg = 30 \, \text{N} - 20 \, \text{N} = 10 \, \text{N}

Using F = ma, we find:

10 \, \text{N} = 2.04 \, \text{kg} \cdot a

Solving for a gives a = 4.9 \, \text{m/s²}, indicating that the book accelerates upwards.

In summary, the normal force is crucial for understanding the balance of forces acting on an object. It varies depending on the applied forces and the weight of the object, and it can be calculated using the principles of equilibrium and Newton's laws of motion. Recognizing these relationships allows for accurate predictions of an object's behavior under various conditions.

Here’s what students ask on this topic:

What is vertical equilibrium in physics?

Vertical equilibrium in physics occurs when the sum of all vertical forces acting on an object is zero, resulting in no vertical acceleration. This means that the object is either at rest or moving with a constant velocity in the vertical direction. Mathematically, this is expressed as:

$\sum F_y=0$

where $F_y$ represents the vertical forces. In practical terms, this often involves balancing the weight force (gravity) with the normal force exerted by a surface.

How do you calculate the normal force?

The normal force is calculated using Newton's second law, $F=ma$. In equilibrium scenarios, the acceleration is zero, so the sum of forces in the vertical direction must be zero. For an object resting on a flat surface, the normal force $N$ balances the weight force $\mathrm{mg}$:

$N=\mathrm{mg}$

where $m$ is the mass of the object and $g$ is the acceleration due to gravity (9.8 m/s²).

What is the difference between equilibrium and normal force?

Equilibrium refers to a state where all forces acting on an object cancel out, resulting in zero acceleration. This can occur in any direction, not just vertically. Normal force, on the other hand, is a specific force that acts perpendicular to the surface of contact between two objects. It is one of the forces that can contribute to equilibrium. For example, in vertical equilibrium, the normal force balances the weight force to ensure no vertical acceleration.

Can an object be in equilibrium if it is moving?

Yes, an object can be in equilibrium while moving. Equilibrium means that the net force acting on the object is zero, resulting in zero acceleration. This implies that the object can move at a constant velocity. For example, a box sliding on a frictionless surface at a constant speed is in equilibrium because the forces acting on it cancel out, resulting in no change in its velocity.

What happens to the normal force if an additional force is applied to an object?

If an additional force is applied to an object, the normal force will adjust to maintain equilibrium. For instance, if a downward force is applied, the normal force will increase to balance the additional force along with the weight. Conversely, if an upward force is applied, the normal force will decrease. The new normal force can be calculated using:

$N=\mathrm{mg}\pm F$

where $F$ is the additional force applied.