7.8
Work-energy Theorem
According to Newton’s second law of motion, the sum of all the forces acting on a particle (net force) determines the rate of change in the momentum o…
The work-energy theorem states that the total work done by all the forces on an object is equal to the change in kinetic energy of that object, where m is the mass and v1 and v2 are the initial and final velocities of the object.
For instance, when a force is applied to a box, the velocity of the box increases, which increases its kinetic energy. Hence, the work done by the force contributes to a change in the kinetic energy of the box.
According to Newton's second law of motion, force is equal to mass times acceleration.
Consider a case where the box is moving with an acceleration a along a straight line. From the third equation of motion, the relationship between velocity and acceleration can be rearranged to obtain the displacement covered by the box.
By substituting the force and displacement terms in the work formula, a relationship between work done and change in kinetic energy is obtained. This expression is called the work-energy theorem.
According to this theorem, when an applied force increases the velocity of an object, the work done is positive. If the applied force causes no change in the velocity of the object or its kinetic energy, the work done is said to be zero.
If the net force applied is used to stop the moving object, the kinetic energy decreases, and hence the work done is negative.
View the full transcript and gain access to JoVE Core videos
Q1: What does the work-energy theorem state?
The work-energy theorem states that the total work done by all forces on an object equals the change in kinetic energy of that object. This relationship connects force, displacement, and velocity through the equation W = ½m(v₂² - v₁²), where m is mass and v₁ and v₂ are initial and final velocities. The theorem provides a powerful tool for analyzing motion when acceleration varies.
Q2: How do you determine if work done is positive, negative, or zero?
Work is positive when an applied force increases an object's velocity and kinetic energy. Work is zero when a force causes no change in velocity or kinetic energy. Work is negative when a net force opposes motion, decreasing kinetic energy. Understanding positive, negative, and zero work helps predict how forces affect an object's motion and energy state.
Q3: How is the work-energy theorem derived from Newton's second law?
Starting with Newton's second law (F = ma), you substitute force and displacement into the work formula. Using the third equation of motion to relate velocity and acceleration, you derive the relationship W = ½m(v₂² - v₁²). This mathematical derivation shows how work done by forces directly produces changes in kinetic energy.
Q4: What steps should you follow when solving work-energy theorem problems?
First, identify given quantities and draw a free-body diagram showing all forces. Calculate work done by each known force, tracking signs carefully. Sum total work from all forces. Apply the work-energy theorem, equating total work to change in kinetic energy. Substitute known values and solve for unknowns. Finally, evaluate your answer for reasonableness.
Q5: Why must you consider net work instead of individual forces?
According to Newton's second law, the net force determines how an object's motion changes. Therefore, only the total work done by all forces affects kinetic energy. Considering work and energy for variable forces requires summing contributions from every force acting on the object to accurately predict energy changes.
Q6: When is the work-energy theorem most useful?
The work-energy theorem is especially valuable when acceleration is not constant. Instead of tracking acceleration throughout motion, you only need initial and final velocities to find work done. This makes the theorem ideal for analyzing complex motion scenarios where forces vary, providing a simpler alternative to kinematic equations.
Q7: What does it mean if total work done on an object is zero?
If total work done is zero, the object's kinetic energy remains unchanged, meaning it travels at constant speed. This occurs when forces are balanced or when applied forces do no net work on the object. Zero total work indicates the object maintains its current velocity without acceleration or deceleration.