Vertical Forces & Acceleration: Videos & Practice Problems

Gravity exerts a downward force known as weight (W or F_g), calculated using the equation W = mg, where m is mass and g is the acceleration due to gravity (approximately 9.8 m/s² on Earth). This force influences an object's motion, leading to acceleration (a). If upward forces exceed downward forces, the object accelerates upward; if downward forces are greater, it accelerates downward. In equilibrium, forces balance, resulting in zero acceleration. Understanding these principles is crucial for solving problems involving vertical forces and motion.

Weight Force & Gravitational Acceleration

In the study of forces, it is essential to understand the role of gravity, particularly the gravitational force, which acts on objects near the Earth. This force, commonly referred to as weight, is a result of the attraction between objects with mass, such as the Earth and an asteroid. The gravitational force can be expressed mathematically as:

$$ W = mg $$

where \( W \) represents the weight force (or gravitational force), \( m \) is the mass of the object, and \( g \) is the acceleration due to gravity. The standard value of \( g \) near the Earth's surface is approximately \( 9.8 \, \text{m/s}^2 \), and it is important to note that weight is measured in newtons (N), not kilograms.

When an object is solely influenced by the weight force, it will experience an acceleration equal to \( g \). This relationship can be derived from Newton's second law of motion, \( F = ma \). If the only force acting on an object is its weight, then:

$$ W = mg = ma $$

By canceling the mass \( m \), we find that the acceleration \( a \) is equal to \( g \). This means that all objects, regardless of their mass, will accelerate towards the Earth at the same rate when only gravity acts upon them.

It is also crucial to differentiate between mass and weight. Mass, which is a measure of the amount of matter in an object, remains constant regardless of location and is measured in kilograms (kg). In contrast, weight varies depending on the gravitational acceleration at a given location. For example, an object with a mass of \( 70 \, \text{kg} \) has a weight of:

$$ W = 70 \, \text{kg} \times 9.8 \, \text{m/s}^2 = 686 \, \text{N} $$

However, if this object were on the Moon, where the gravitational acceleration is approximately \( 1.62 \, \text{m/s}^2 \), its weight would be:

$$ W = 70 \, \text{kg} \times 1.62 \, \text{m/s}^2 = 113.4 \, \text{N} $$

This illustrates that while the mass remains \( 70 \, \text{kg} \) in both locations, the weight changes due to the difference in gravitational acceleration.

In summary, understanding the distinction between mass and weight, as well as the implications of gravitational acceleration, is fundamental in physics. The weight force always acts towards the center of the Earth, and its calculation is vital for solving problems related to motion and forces in various contexts.

The Mars Rover Perseverance weighed about 10,000 N while on Earth. After it reached the surface of Mars, it weighed about 3790 N. What is the gravitational acceleration on Mars?

• A. $g_{Mars}=0.039m/s^2$
• B. $g_{Mars}=0.27m/s^2$
• C. $g_{Mars}=3.7m/s^2$
• D. $g_{Mars}=26m/s^2$

Answer: (C)

Vertical Forces And Acceleration in the Y-axis

Understanding the dynamics of forces acting on an object is crucial in physics, particularly when analyzing situations where vertical forces do not cancel out. In such cases, the object will experience acceleration in the vertical direction. To solve these problems, we can follow a systematic approach using free body diagrams and Newton's second law of motion, which states that the net force acting on an object is equal to the mass of the object multiplied by its acceleration, expressed as F = ma.

Consider a scenario with a 5.1 kg block suspended in the air by a massless string. The weight force acting on the block, which is the force due to gravity, can be calculated using the formula w = mg, where g is the acceleration due to gravity (approximately 9.8 m/s²). Thus, the weight force is:

w = -mg = -5.1 kg × 9.8 m/s² = -50 N

When a tension force is applied, we can analyze the system by drawing a free body diagram. For example, if the tension is 70 N, the net force can be calculated as:

F_net = T - w = 70 N - 50 N = 20 N

Applying Newton's second law, we have:

F_net = ma

Substituting the values gives:

20 N = 5.1 kg × a

Solving for acceleration results in:

a = 3.92 m/s²

This positive acceleration indicates that the block accelerates upwards, as the tension force exceeds the weight force.

In contrast, if the tension is reduced to 30 N, the net force becomes:

F_net = T - w = 30 N - 50 N = -20 N

Using F = ma again, we find:

-20 N = 5.1 kg × a

Thus, the acceleration is:

a = -3.92 m/s²

The negative sign indicates that the block accelerates downwards, as the weight force is greater than the tension force.

When the tension equals the weight force (50 N), the forces balance out:

F_net = T - w = 50 N - 50 N = 0 N

In this case, the acceleration is:

a = 0 m/s²

This situation represents equilibrium, where the object remains at rest or moves with constant velocity.

Finally, if there is no tension (0 N), the only force acting on the block is its weight:

F_net = -50 N

Applying F = ma gives:

-50 N = 5.1 kg × a

Solving for acceleration yields:

a = -9.8 m/s²

This result confirms that the block accelerates downwards at the acceleration due to gravity.

In summary, the relationship between tension and weight forces determines the direction and magnitude of acceleration. If the upward forces exceed the downward forces, the object accelerates upwards. Conversely, if the downward forces are greater, the object accelerates downwards. When the forces are equal, the object is in equilibrium, resulting in zero acceleration. Lastly, if there are no upward forces, the object accelerates downwards at g.

A 3-kg bucket is being pulled upwards by a cord. The tension in the cord is 35 N. What is the acceleration of the bucket? (The mass of the cord is negligible, which means you can assume m_cord = 0.)

• A. 25 m/s²
• B. 1.9 m/s²
• C. 21 m/s²
• D. 8.4 m/s²

Answer: (B)

Lowering a Load of Bricks

In this problem, we analyze a 100-kilogram load of bricks being lowered by a cable at a velocity of 5 meters per second. The load eventually comes to a stop over a time period of 2 seconds. To determine the tension in the cable required to achieve this, we first establish the initial and final velocities: the initial velocity (\(v_0\)) is -5 m/s (downward), and the final velocity (\(v\)) is 0 m/s (at rest).

To find the acceleration (\(a\)), we can use the kinematic equation:

\[ v = v_0 + a \cdot t \]

Substituting the known values, we have:

\[ 0 = -5 + a \cdot 2 \]

Rearranging gives:

\[ 5 = 2a \implies a = 2.5 \, \text{m/s}^2 \

This positive acceleration indicates that the direction of acceleration is upward, which is consistent with the need to stop the downward motion of the bricks.

Next, we draw a free body diagram to analyze the forces acting on the load. The weight force (\(W\)) acting downward is given by:

\[ W = mg = 100 \, \text{kg} \cdot 9.8 \, \text{m/s}^2 = 980 \, \text{N} \]

The tension force (\(T\)) in the cable acts upward. According to Newton's second law, the net force (\(F\)) can be expressed as:

\[ F = T - W = ma \]

Substituting the expressions for weight and acceleration, we have:

\[ T - 980 = 100 \cdot 2.5 \]

Solving for tension gives:

\[ T = 980 + 250 = 1230 \, \text{N} \

This result indicates that the tension in the cable must be 1230 Newtons to ensure the load of bricks decelerates to a stop. This value is greater than the weight force, which is necessary to produce the upward acceleration required to halt the downward motion of the bricks.

Here’s what students ask on this topic:

What is the difference between weight and mass?

Weight and mass are often confused but are fundamentally different. Mass is a measure of the amount of matter in an object and is measured in kilograms (kg). It remains constant regardless of location. Weight, on the other hand, is the force exerted by gravity on that mass. It is calculated using the equation $W=mg$, where $W$ is weight, $m$ is mass, and $g$ is the acceleration due to gravity (approximately 9.8 m/s² on Earth). Weight is measured in newtons (N) and varies with the gravitational field strength of the location.

How do you calculate the weight of an object on Earth?

To calculate the weight of an object on Earth, you use the equation $W=mg$, where $W$ is the weight, $m$ is the mass of the object in kilograms (kg), and $g$ is the acceleration due to gravity, which is approximately 9.8 m/s² on Earth. For example, if an object has a mass of 10 kg, its weight would be $10\times 9.8=98$ newtons (N).

What happens to the weight of an object if it is taken to the moon?

The weight of an object decreases if it is taken to the moon because the moon's gravitational acceleration is weaker than Earth's. The gravitational acceleration on the moon is approximately 1.62 m/s². Using the weight equation $W=mg$, if an object has a mass of 10 kg, its weight on the moon would be $10\times 1.62=16.2$ newtons (N), compared to $98$ newtons on Earth.

How do you determine the acceleration of an object when vertical forces do not cancel out?

To determine the acceleration of an object when vertical forces do not cancel out, use Newton's second law, $F=ma$. First, draw a free-body diagram to identify all forces acting on the object. Sum the forces to find the net force $F$_net. Then, use the equation $F$_net=ma to solve for acceleration $a$. For example, if a 5.1 kg block has a tension force of 70 N upwards and a weight force of 50 N downwards, the net force is $70-50=20$ N. Using $20=5.1a$, the acceleration $a$ is approximately 3.92 m/s² upwards.

What is gravitational acceleration and how does it vary by location?

Gravitational acceleration, denoted as $g$, is the acceleration due to the force of gravity acting on an object. On Earth, it is approximately 9.8 m/s². However, this value varies depending on the location. For example, on the moon, gravitational acceleration is about 1.62 m/s², and on Jupiter, it is approximately 24.79 m/s². These variations occur because different celestial bodies have different masses and radii, affecting the strength of their gravitational fields.