Question

A cart is pulled a distance d on a horizontal surface while it is acted on by five forces, as shown in the diagram. Normal forces of magnitudes n1 and n2 are directed upwards at the rear and front wheels, respectively. Rolling frictional forces of magnitudes f1 and f2 are opposite in direction to the displacement and act on the rear and front wheels, respectively. The weight of the cart, with magnitude w, is directed downward. Finally, a pulling force of magnitude p makes and angle of φ with horizontal. The direction of the displacement is to the right in the diagram.

Answer 1

The work done by the pulling force is $P\cdot d \cdot \cos \theta$.

The work done by rear wheels normal force is 0.

The work done by front wheels normal force is 0.

The work done by rear wheels friction force is $-f_{1}\cdot d$.

The work done by front wheels friction force is $-f_{2}\cdot d$.

The work done by weight is 0.

How to determine the work done by a force

In this question, we see a system with six uniform and stable forces. The equation of work done by a uniform and stable force is:

$W = F\cdot s\cdot \cos \theta$ (1)

Where:

• $F$ - Force, in newtons.
• $s$ - Traveled distance, in meters.
• $\theta$ - Angle of force respect to displacement direction, in degrees.

Now we proceed to calculate the work done by each force:

Pulling force ($P$)

$W_{P} = P\cdot d\cdot \cos \theta$ (2)

The work done by the pulling force is $P\cdot d \cdot \cos \theta$. $\blacksquare$

Rear wheels normal force ($N_{1}$)

$W_{N_{1}} = N_{1}\cdot d\cdot \cos 90^{\circ}$

$W_{N_{1}} = 0$ (3)

The work done by rear wheels normal force is 0. $\blacksquare$

Front wheels normal force ($N_{2}$)

$W_{N_{2}} = N_{2}\cdot d\cdot \cos 90^{\circ}$

$W_{N_{2}} = 0$ (4)

The work done by front wheels normal force is 0. $\blacksquare$

Rear wheels friction force ($f_{1}$)

$W_{f_{1}} = f_{1}\cdot d\cdot \cos 180^{\circ}$

$W_{f_{1}} = - f_{1}\cdot d$ (5)

The work done by rear wheels friction force is $-f_{1}\cdot d$. $\blacksquare$

Front wheels friction force ($f_{2}$)

$W_{f_{2}} = f_{2}\cdot d\cdot \cos 180^{\circ}$

$W_{f_{2}} = -f_{2}\cdot d$ (6)

The work done by front wheels friction force is $-f_{2}\cdot d$. $\blacksquare$

Weight ($W$)

$W_{W} = W\cdot d\cdot \cos 270^{\circ}$

$W_{W} = 0$ (7)

The work done by weight is 0. $\blacksquare$

Remark

The question is poorly formatted and a figure is missing. Correct form is shown below:

A cart is pulled a distance $d$ on a horizontal surface while it is acted on by forces, as shown in the diagram. What is the work done by each force?

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