Net Work & Work-Energy Theorem: Videos & Practice Problems

Understanding net work involves calculating the total work done by multiple forces acting on an object. The net work is the sum of individual works, calculated using the equation ${\frac{F}{d}}^{\cos }\left(\theta \right)$. The work-energy theorem relates net work to the change in kinetic energy, expressed as $W=\Delta K=K\_\mathrm{final}-K\_\mathrm{initial}$. This relationship allows for the calculation of work even when forces are unknown, emphasizing the connection between work and energy transfer.

Calculating Net Work

When calculating the net work done on an object by multiple forces, the key concept is that the total or net work is the sum of the work done by each individual force. This net work can be found using two main approaches depending on the information given and what the problem asks for.

The first method involves calculating the work done by each force separately using the formula for work:

\[ W = F \cdot d \cdot \cos \theta \]

where F is the magnitude of the force, d is the displacement, and θ is the angle between the force and displacement vectors. For example, if a force acts in the same direction as the displacement, the angle is 0°, and the cosine term is 1, simplifying the work to force times distance. Forces acting opposite to the displacement, such as friction, produce negative work because the angle is 180°, and cosine of 180° is -1. Forces perpendicular to the displacement, like the normal force or weight when motion is horizontal, do no work since the cosine of 90° is zero.

After calculating the work done by each force, the net work is simply the algebraic sum of all these individual works:

\[ W_{\text{net}} = \sum W_i \]

This approach is straightforward when the problem provides all forces and their directions explicitly.

The second method starts by finding the net force acting on the object. This involves summing all forces vectorially, often by separating them into components along the axes. For horizontal motion, vertical forces like weight and normal force cancel out, so only horizontal forces contribute to the net force:

\[ F_{\text{net}} = \sum F_x \]

Once the net force is determined, the net work can be calculated using the same work formula, treating the net force as a single force:

\[ W_{\text{net}} = F_{\text{net}} \cdot d \cdot \cos \theta \]

where θ is the angle between the net force and displacement vectors. If they are parallel, cosine of 0° equals 1, simplifying the calculation.

Both methods yield the same result because calculating work for each force and then summing is mathematically equivalent to summing forces first and then calculating work. The choice of method depends on the problem context and personal preference. Understanding these approaches enhances problem-solving flexibility in physics, especially in work and energy calculations involving multiple forces.

You pull a 3kg box on a flat surface. The coefficient of kinetic friction is 0.6. When you pull the box horizontally through a distance of 10m, it accelerates at 2m/s². Find the net work on the box.

• A. 17.6 J
• B. 35.3 J
• C. 176.4 J
• D. 60 J

Answer: (D)

To pull a 51 kg crate across a rough floor, a worker applies a force of 100 N, directed 37°above the horizontal. The coefficient of friction is 0.16. If the crate moves 3.0 m, what is the total work done on the crate?

• A. 29 J
• B. 0 J
• C. 240 J
• D. 169.6 J

Answer: (A)

The Work-Energy Theorem

Work is defined as the amount of energy transferred between objects, often resulting in a change in an object's kinetic energy. For example, if you push a book and it gains 10 joules of kinetic energy, you have done 10 joules of work on the book. This concept is mathematically expressed through the work-energy theorem, which establishes a direct relationship between the net work done on an object and its change in kinetic energy.

The work-energy theorem states that the net work (\(W_{\text{net}}\)) done by all forces acting on an object equals the change in the object's kinetic energy (\(\Delta K\)), which is the difference between the final kinetic energy (\(K_{\text{final}}\)) and the initial kinetic energy (\(K_{\text{initial}}\)):

\[W_{\text{net}} = K_{\text{final}} - K_{\text{initial}}\]

Kinetic energy (\(K\)) itself is calculated using the formula:

\[K = \frac{1}{2} m v^2\]

where m is the mass of the object and v is its speed. This theorem is particularly useful when the forces acting on an object are unknown or unspecified, as it allows the calculation of net work solely from changes in speed and mass.

For instance, consider a 4-kilogram box that moves from an initial speed of 6 m/s to a final speed of 10 m/s. To find the net work done on the box, apply the work-energy theorem by calculating the difference in kinetic energy at these two speeds:

\[W_{\text{net}} = \frac{1}{2} m v_{\text{final}}^2 - \frac{1}{2} m v_{\text{initial}}^2\]\[W_{\text{net}} = \frac{1}{2} \times 4 \times 10^2 - \frac{1}{2} \times 4 \times 6^2 = 2 \times 100 - 2 \times 36 = 200 - 72 = 128 \text{ joules}\]

This calculation shows that 128 joules of net work were done on the box to increase its speed from 6 m/s to 10 m/s. The work-energy theorem thus provides a powerful tool for analyzing energy transfer without needing detailed information about the forces involved or the direction of motion. Understanding this relationship enhances problem-solving skills in physics by linking energy concepts with mechanical work in a straightforward and efficient manner.

A box slides across the floor with an initial speed of 3.5m/s. If the coefficient of kinetic friction is 0.15, how far will the box slide before stopping completely?

• A. 1.19 m
• B. 4.17 m
• C. 9 m
• D. Not Enough Information

Answer: (B)

Net Work & Constant Speed

The Work Energy Theorem is a fundamental concept in physics that relates the work done on an object to its change in kinetic energy. In scenarios where an object moves at a constant speed, such as a 5-kilogram box being pushed on a rough surface, understanding this theorem can simplify problem-solving significantly.

In this example, the box is pushed with an applied force while experiencing kinetic friction. Since the box moves at a constant speed of 8 meters per second, it is crucial to recognize that the acceleration is zero. According to Newton's second law, the net force acting on the box can be expressed as:

\[ F_{\text{net}} = m \cdot a \]

Given that the acceleration (a) is zero, the net force (F_net) must also be zero. This indicates that the applied force is balanced by the force of friction, resulting in no net force acting on the box.

To calculate the net work done on the box, we can use the equation:

\[ W_{\text{net}} = F_{\text{net}} \cdot d \cdot \cos(\theta) \]

Substituting the net force of zero into this equation yields:

\[ W_{\text{net}} = 0 \cdot d \cdot \cos(\theta) = 0 \text{ joules} \]

This outcome illustrates that if the net work done on an object is zero, then the change in kinetic energy is also zero, as stated by the Work Energy Theorem. The kinetic energy (KE) of an object is defined as:

\[ KE = \frac{1}{2} m v^2 \]

Since the speed (v) remains constant, the kinetic energy does not change, confirming that the net work done is zero. This principle is particularly useful in problem-solving; if you encounter a situation where an object is moving at a constant speed, you can immediately conclude that the net work done on the object is zero, and thus, there is no change in kinetic energy.

In summary, the Work Energy Theorem provides a powerful tool for analyzing motion. Recognizing that constant speed implies zero net work and no change in kinetic energy can streamline your approach to various physics problems.

Here’s what students ask on this topic:

What is the net work done on an object?

The net work done on an object is the total work resulting from all the forces acting on it. It is calculated by summing the individual works done by each force. Mathematically, it can be expressed as:

$W=\sum _{}^{i}W\_i$

where $W\_i$ is the work done by the $i$-th force. Each work can be calculated using the formula:

$W=Fdcos\theta$

where $F$ is the force, $d$ is the displacement, and $\theta$ is the angle between the force and displacement vectors.

How do you calculate net work using the work-energy theorem?

The work-energy theorem states that the net work done on an object is equal to the change in its kinetic energy. This can be expressed as:

$W=\Delta K=K\_f-K\_i$

where $K\_f$ is the final kinetic energy and $K\_i$ is the initial kinetic energy. Kinetic energy is given by:

$K=\frac{1}{2}m{v}^2$

where $m$ is the mass and $v$ is the velocity. By knowing the initial and final velocities, you can calculate the change in kinetic energy and thus the net work done.

What is the relationship between work and energy transfer?

Work is a measure of energy transfer between objects. When work is done on an object, energy is transferred to or from the object. For example, if you push a book and it gains speed, the work you do on the book results in an increase in its kinetic energy. The relationship between work and energy is given by the work-energy theorem:

$W=\Delta K$

This equation shows that the net work done on an object is equal to the change in its kinetic energy, emphasizing the direct connection between work and energy transfer.

How do you calculate work done by multiple forces?

To calculate the work done by multiple forces, you need to determine the work done by each individual force and then sum them up. The work done by a single force is given by:

$W=Fdcos\theta$

where $F$ is the force, $d$ is the displacement, and $\theta$ is the angle between the force and displacement vectors. After calculating the work for each force, sum them up to get the net work:

$W=\sum _{}^{i}W\_i$

What is the work-energy theorem?

The work-energy theorem is a fundamental principle in physics that relates the net work done on an object to its change in kinetic energy. It states that:

$W=\Delta K=K\_f-K\_i$

where $K\_f$ is the final kinetic energy and $K\_i$ is the initial kinetic energy. This theorem highlights the direct relationship between the work done on an object and the resulting change in its kinetic energy, providing a powerful tool for solving problems involving energy transfer.