Interactive physics simulator
Positive Work
Explore how forces that act in the direction of motion perform positive work, transferring energy to an object and causing it to speed up.
Positive Work Lab
Analyze the relationship between force angle, displacement, and kinetic energy gain in real time.
Live Telemetry
- Applied Force (F)
- 50.0 N
- Force Angle (θ)
- 30°
- Work Done (W)
- 0.0 J
- Kinetic Energy (KE)
- 0.0 J
Understanding Positive Work
In physics, positive work represents a transfer of energy into an object. This occurs when the force acting on the object has a component pointing in the same direction as its displacement.
Mathematically, positive work is defined through the dot product of the force and displacement vectors:
where:
- F is the magnitude of the applied force (Newtons, N).
- s is the displacement of the object (meters, s).
- θ is the angle between the force vector and the displacement vector.
For work to be positive, the term cos(θ) must be greater than zero, which corresponds to an acute angle range of:
Key Concept: Energy Transfer
According to the Work-Energy Theorem (Wnet = Δ KE), doing positive net work on an object always increases its kinetic energy. Because kinetic energy is proportional to velocity squared (KE = (1)/(2)mv2), doing positive work causes the object to speed up. The object acts as an energy receiver, absorbing mechanical work and converting it into translational motion.
Constant vs Variable Positive Work
Calculating positive work depends on whether the assistive force remains constant or changes over the distance:
- Constant Pull/Push: Work done is a simple product W = (F cosθ) · s. On a Force-Displacement graph, this is represented by a rectangular area.
- Variable Force (Bow/Spring): As a drawn bowstring is released, the force varies. The work done equals the area under the force-displacement slope, which forms a triangle: W = (1)/(2) Fmax · x.
Solved Examples
A dog pulls a sled forward across flat snow with a horizontal tension force of F = 35 N. The sled travels a displacement of s = 15 meters in the exact direction of the pull. Calculate the work done by the dog and explain why it is positive.
- Identify the given values: Pulling force F = 35 N, displacement s = 15 m.
- Identify the angle θ between the force and displacement: Since they are in the exact same direction, θ = 0° and cos(0°) = 1.
- Recall the simplified work equation: W = F · s · cos(θ).
- Substitute values: W = 35 N · 15 m · 1 = 525 Joules.
- Explain the sign: Because the force acts in the same direction as the motion (θ = 0°), work is positive (W > 0). This work transfers 525 J of energy to the sled, increasing its kinetic energy.
Answer: Work Done W = +525 J
A cargo crate of mass m = 8 kg is dropped from a crane and falls vertically downwards through a height of h = 6 meters. Calculate the work done on the crate by gravity. (Use gravity acceleration g = 9.8 m/s²).
- Identify the given values: Mass m = 8 kg, height h = 6 m, gravity acceleration g = 9.8 m/s².
- Calculate the force of gravity acting on the crate: Fg = m · g = 8 kg · 9.8 m/s² = 78.4 N (acting downwards).
- Identify the displacement direction: The crate falls downwards, so displacement s = 6 m is downwards.
- Determine the angle θ: Both the force of gravity and the displacement point straight down, so the angle θ = 0° and cos(θ) = 1.
- Calculate gravity work: W = Fg · s · cos(0°) = m · g · h = 78.4 N · 6 m = 470.4 Joules.
- Since the gravitational pull assists the downward fall, gravity does +470.4 J of positive work, converting potential energy into kinetic energy (making the crate speed up).
Answer: Work Done W = +470.4 J
An archer draws a bowstring back by a distance of x = 0.7 meters with a maximum force of Fmax = 120 N. Assuming the elastic force of the bow string varies linearly (Hooke's Law) as it pushes the arrow forward, calculate the work done by the bow on the arrow upon release.
- Identify the variables: Draw distance x = 0.7 m, maximum force Fmax = 120 N.
- Understand the force profile: The spring/tension force starts at 120 N and decreases linearly to 0 N as the string relaxes. This is a variable force.
- Recall the formula for positive work done by a spring/elastic force: W = 1/2 · Fmax · x.
- Substitute the values: W = 1/2 · 120 N · 0.7 m.
- Calculate the result: W = 60 · 0.7 = 42 Joules.
- The bow does +42 J of positive work on the arrow, accelerating it to launch speed.
Answer: Work Done W = +42 J
Common Mistakes
- Assuming all forces do positive work. Forces opposing motion (like friction, θ = 180^°) do negative work, extracting energy from the object.
- Confusing vertical normal forces with work. A normal force acts perpendicular to a horizontal surface (θ = 90^°). Since cos(90^°) = 0, it does exactly zero work.
- Ignoring displacement boundaries. If a force is applied but the object does not move (s = 0), the work done is exactly zero, regardless of the force magnitude.
Gravity Assist Physics
When an object is dropped, the downward force of gravity acts in the same direction as the downward displacement (s = h, θ = 0^°). Gravity does positive work on the object:
This positive work converts gravitational potential energy into kinetic energy, accelerating the object downward.
Practice Questions
1. A passenger pulls a rolling suitcase with a force of F = 40 N at an angle of θ = 45° above the horizontal, moving it a distance of s = 12 meters. Does this force do positive work, and what is its value?
Yes, it does positive work because the angle θ = 45° is acute (cos(45°) = 0.7071 > 0). The work done is W = F · s · cos(θ) = 40 · 12 · cos(45°) = 480 · 0.7071 ≈ +339.4 J. The horizontal component of the force pulls the suitcase forward, increasing its energy.
2. When a car accelerates forward, what force performs positive work on the car to increase its kinetic energy?
The static friction force exerted by the road surface forward on the car's drive tires performs the positive work. While it may seem counterintuitive, this friction force is what prevents the wheels from slipping and pushes the vehicle forward in the direction of displacement.
3. Why does the normal force on a block sliding down a ramp do zero work, while gravity does positive work?
The normal force acts perpendicular to the surface of the ramp (at an angle of 90° to the sliding displacement), so its work is W = N · s · cos(90°) = 0 J. Gravity acts straight down, which has an acute angle component along the ramp slope, doing positive work (W = mg · s · sin(α) > 0) that speeds up the block.
4. What happens to the velocity of an object if the net work done on it is positive?
According to the Work-Energy Theorem (W_net = ΔKE), if the net work is positive, the object's change in kinetic energy is positive (ΔKE > 0). This means the object's final kinetic energy is greater than its initial kinetic energy, which causes the velocity of the object to increase.
FAQ
Frequently Asked Questions
What is positive work in physics?
Positive work is done when the applied force has a component in the same direction as the displacement of the object, transferring energy to the object to speed it up. Mathematically, this occurs when the angle θ between the force and displacement vectors is acute (-90° < θ < 90°), meaning cos(θ) > 0.
What is the mathematical condition for positive work?
The mathematical condition is that the angle θ between the force vector and the displacement vector must satisfy -90° < θ < 90°. In this range, cos(θ) is positive, which makes the work done (W = F · s · cos(θ)) positive.
What is the effect of positive work on an object's energy?
According to the Work-Energy Theorem, positive work done on an object increases its kinetic energy. This energy transfer causes the object to speed up (velocity increases).
What are some common real-life examples of positive work?
Examples include pulling a sled forward (force assists motion), a crane lifting a cargo load vertically upward (upward force and upward displacement), gravity pulling a falling skydiver down, and a bowstring accelerating an arrow forward.
Does gravity perform positive work?
Gravity performs positive work when an object moves downward (since both gravity force and displacement are downward, θ = 0°). When an object is thrown upward, gravity opposes the motion (θ = 180°), doing negative work.
Can a force acting at an angle still perform positive work?
Yes. As long as the angle θ is less than 90° (an acute angle), the parallel component of the force (F_parallel = F cos(θ)) is positive, which does positive work along the displacement direction.
What is the difference between positive work and negative work?
Positive work adds kinetic energy to an object, helping or driving the motion (θ < 90°). Negative work removes kinetic energy, acting as resistance or friction opposing the motion (θ > 90°).
What is the work done by a spring when it accelerates a box forward?
When a stretched spring contracts and pulls a block back toward its relaxed state, the spring force acts in the direction of the block's displacement, doing positive work and transferring stored elastic potential energy into kinetic energy.
Is work positive when you lift a heavy backpack?
Yes. The lifting force you exert is upward, and the displacement is upward, so your force performs positive work on the backpack. However, the force of gravity performs negative work in this scenario because gravity pulls downward while displacement is upward.
How does the Work-Energy Theorem relate to positive work?
The Work-Energy Theorem states that net work done equals the change in kinetic energy (W_net = ΔKE). If the net work is positive, the change in kinetic energy is positive (ΔKE > 0), resulting in an increase in the object's speed.
What is the SI unit of positive work?
Like all forms of work, energy, and heat, the SI unit of positive work is the Joule (J), which is equivalent to one Newton-meter (1 N·m).
Why does friction usually not do positive work?
Friction is a resistive force that opposes the direction of relative sliding motion. Because the friction force vector points in the opposite direction of displacement (θ = 180°), cos(180°) = -1, which results in negative work.