Date: ______________________

SPH4U: Energy and Momentum

Introduction to Work

In physics, work is defined as a physical quantity relating the size and direction of a force acting on an

object causing the object to experience a displacement. This is a directional relationship between the

force vector and the displacement vector. The scalar quantity of work can be evaluated based on the dot

product between these two vectors.

𝑊 = �⃗� ∙ Δ𝑑⃗⃗⃗⃗⃗⃗ 𝑜𝑟 𝑊 = |�⃗�||Δ𝑑|𝑐𝑜𝑠𝜃

Where the angle (𝜃) is measured between the tails of the two vectors.

Note: The sign of a dot product can be determined by the angle between the two vectors.

We can think of positive work as an increasing energy of an object, while negative work removes energy

from an object. Consider this by analyzing the following example of a cart moving through a

displacement of 10.0 m.

Determine the work done by the applied force: Determine the work done by the gravitational force:

Determine the work done by the normal force:

Determine the work done by the frictional force:

Thus, the overall work done on the cart is:

Equally the total work done on the object can be found using the equation: 𝑊 = |�⃗�𝑛𝑒𝑡||Δ𝑑|𝑐𝑜𝑠𝜃

Date: ______________________

SPH4U: Energy and Momentum

Notice that the example above dealt with constant forces. When we have forces acting on the object

change, we have two different approaches.

1. Area under the curve.

Work is represented by the area under the curve on a force versus displacement graph.

2. Utilizing the average force.

Consider lifting a 10.0 kg mass through a height of 1.00 m. Initially a force greater than

the force of gravity must be applied to the mass to get it moving. During most of the lift, the object will

travel at a constant speed. During this period, the applied force will be equal and opposite to gravity. To

bring the mass to rest (1.00 m higher), the applied force must be less than the force of gravity. On

AVERAGE during the lift the force required is equal and opposite to the force of gravity acting on the

mass. The speed of your motion will not effect this analysis.

Therefore, the work done by the applied force is:

𝑊 = |𝐹𝑔⃗⃗ ⃗⃗ ||Δ𝑑| cos(𝜃)

= (𝑚𝑔)(Δ𝑑) cos(𝜃)

= (10.0𝑘𝑔) (9.8𝐾

𝑘𝑔) (1.0 𝑚) cos(0°)

= 98 𝐽

Gravity also does work (on our muscles)

𝑊 = |𝐹𝑔⃗⃗ ⃗⃗ ||Δ𝑑| cos(𝜃)

= (𝑚𝑔)(Δ𝑑) cos(𝜃)

= (10.0𝑘𝑔) (9.8𝐾

𝑘𝑔) (1.0 𝑚) cos(180°)

= −98 𝐽

This is an example of the law of conservation of energy because as the mass gains energy, we lose it!