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Wize University Physics Textbook (Master) > Work and Energy

Mechanical Work (Basic)

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Mechanical Work

Mechanical work is defined as the dot product of displacement vector and force vector. For a constant force applied over the object in Δr⃗\Delta\vec{r} distance, the work done by the force is:

W=F⃗. Δr⃗ =FΔrcos⁡(θ)\boxed{W=\vec{F}.\ \Delta\vec{r}\ =F\Delta r \cos(\theta)}




where θ\thetais the angle between force direction and direction of motion of the object.
The SI unit of work is the Joule.1 J=1N⋅m=1 kg⋅m2s21\ J=1N\cdot m=1\ \frac{kg\cdot m^2}{s^2}

Wize Tip
Only component of the force along the motion can do a non-zero work on the abject. That is why sometimes we can simply write down mechanical work as W=F∥dW = F_\parallel dwhere F∥ F_\parallel is the component of the force parallel to direction of motion.


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Mechanical Work Sign

Work is a scalar quantity and can be either positive, negative or zero:
  1. If the applied force F (and the displacement d are in the same direction, work done is positive – e.g: pushing block forward
  2. If F and d are in opposite directions (cos 180° = −1), work done is negative – e.g. applying brakes in a moving car
  3. Work is zero if either
  • displacement is zero, so the work done is zero – e.g. pushing against a wall OR
  • θ = 90°, so cos θ = 0 (force is perpendicular to displacement) – e.g. you are carrying a box and moving horizontally (no work done against the force of gravity)

Three cases:








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Example: Sign of Mechanical Work



Which one of the four forces does the most negative work on a particle that undergoes a displacement of Δr⃗\Delta\vec{r}?

Solution:

Note that all cases has the same displacement vector. Using the definition of work as W=F . Δr⃗=F Δrcos⁡(θ)W=F\ .\ \Delta\vec{r}=F\ \Delta r\cos\left(\theta\right) indicates that:

For the top left case:
W=2×Δr×cos⁡(180)=−2ΔrW=2\times\Delta r\times\cos\left(180\right)=-2\Delta r
For the top right case:
W=6×Δr×cos⁡(90)=0W=6\times\Delta r\times\cos\left(90\right)=0
For the bottom left case:
W=5×Δr×cos⁡(50)=+3.2ΔrW=5\times\Delta r\times\cos\left(50\right)=+3.2\Delta r
For the bottom right case:
W=4×Δr×cos⁡(135)=−2.82ΔrW=4\times\Delta r\times\cos\left(135\right)=-2.82\Delta r

Thus, the answer is the 4 N force, on the bottom right corner, which makes angle of 135 degrees with the displacement vector.




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A 20 kg wagon rests on an inclined plane with angle 30°. The wagon is pulled up the inclined plane by a 200 N tension force along the plane and is displaced by 2 m. The coefficient of kinetic friction between the surfaces is 0.3. What is the net work done on this box? (Hint: You can find the net work by finding the work done on it by all the forces acting on it)
Extra Practice
work done
A 70kg70kgman carries a 15kg15kgbox up a flight of stairs. If the angle of the stairs is 25°25\degreeand the man travels a total distance of 14m14m, how much work does he do to the box? How much work does he do in total?
Practice: Work done by gravity
When would the work done by gravity on a simple pendulum be negative?
Forces: Work- wagon on slope
A 20kg wagon rests on an inclined plane with angle 30°. The wagon is pulled by a 200N tension force along the plane and is displaced by 2 m. The coefficient of kinetic friction between the surfaces is 0.30.
A 70kg70kgman carries a 15kg15kgbox up a flight of stairs.
a) If the angle of the stairs is 25°25\degreeand the man travels a total distance of 14m14m, how much work does he do to the box?
b) How much work does he do in total?
What is the work done by tension on the mass in a simple pendulum?
When would the work done by gravity on a simple pendulum be negative?
Consider a simple pendulum that is just beginning to fall after reaching one of its highest positions. In this moment, is the work done by gravity positive, negative, or zero?
An object moving along x-axis is acted upon by a variable force F(x)F\left(x\right) . How much work does the force do on the object as the object moves from x=0 to x=16 m?
You're running up the stairs because you are late for class. Which of the following statements about the work done by gravity on you?
Tom is holding a book and moving it horizontally. What is true about the work Tom is performing on the book?
Work on wagon
A 20kg wagon rests on an inclined plane with angle 30°. The wagon is pulled by a 200N tension force along the plane and is displaced by 2 m. The coefficient of kinetic friction between the surfaces is 0.30.
a) Draw a free-body diagram for the wagon.
b) Find the work done by each force on the wagon.
work done by force of friction
A 5 kg block is moved up a 30 degree incline by a force of 50 N, parallel to the incline. The coefficient of kinetic friction between the block and the incline is .25. How much work is done by the 50 N force in moving the block a distance of 10 meters? What is the total work done on the block over the same distance?
work done
A ball is connected to a rope and swung around in uniform circular motion. The tension in the rope is measured at 10 N and the radius of the circle is 1 m. How much work is done in one revolution around the circle?
work and power
Someone takes 11 minutes to walk up a hill 120m high. His weight is 550N.
(a) How much work did he do in climbing the hill?
(b) What power is developed by his muscles as he climbs?
Work done
A person is holding a bucket by applying a force of 10N. He moves a horizontal distance of 5m and then climbs up a vertical distance of 10m. Find the total work done by him?
Work done by friction
When a body slides against a rough horizontal surface, the work done by friction is ___________
Work done
When a body falls freely under gravity, then the work done by the gravity is ___________
work done
A 70kg70kgman carries a 15kg15kgbox up a flight of stairs. If the angle of the stairs is 25°25\degreeand the man travels a total distance of 14m14m, how much work does he do to the box? How much work does he do in total?
A 100-grams steel ball is revolving around a vertical post by a string, at constant pace of once every second. The circle has radius 30 cm. Which statement(s) is correct for one full revolution?