Question

You slide a 0.12-kg coffee mug $0.15\mathrm{m}$ across a table. The force you exert is horizontal and of magnitude $0.10\mathrm{N}$. The coefficient of kinetic friction between the mug and the table is $0.05$. How much work is done on the mug?

Short answer

The work done on the mug is 0.00617 J.

Step-by-step solution

Understand the Problem

We need to find the work done on the coffee mug as it slides across the table. Work is defined as the force exerted on an object times the distance the object moves in the direction of the force. Mathematically, $W=F\cdot d\cdot \cos (\theta )$ where $F$ is the force, $d$ is the distance, and $\theta$ is the angle between the force and the motion's direction.

Calculate the Net Force

First, we need to determine the net force acting on the mug. The net force was exerted minus kinetic friction force. The force of friction, $F_f$, is given by $F_f={\mu }_k\cdot N$, where ${\mu }_k$ is the coefficient of kinetic friction, and $N$ is the normal force. Since the mug is on a horizontal surface, $N=mg$ (weight of the mug).

Calculate Normal Force

The normal force $N$ is calculated as following:$$N=mg=0.12\text{kg}\times 9.81{\text{m/s}}^2=1.1772\text{N}$$

Calculate Friction Force

Now, calculate the friction force, $F_f$ using:$$F_f={\mu }_k\cdot N=0.05\times 1.1772=0.05886\text{N}$$

Calculate Net Force

The net force, $F_{net}$, is the applied force minus the friction force:$$F_{net}=F-F_f=0.10\text{N}-0.05886\text{N}=0.04114\text{N}$$

Calculate the Work Done

With the net force calculated, now use the work formula. Since the force direction and motion direction are the same, $\theta =0$ degrees. Thus, $\cos (\theta )=1$:$$W=F_{net}\times d\times \cos (0)=0.04114\text{N}\times 0.15\text{m}\times 1=0.006171\text{J}$$

Key concepts

Kinetic Friction

When an object slides across a surface, kinetic friction is the force opposing the motion. It's what slows down the coffee mug as it moves across the table. Kinetic friction depends on two things:

• The nature of the surfaces in contact, represented by the coefficient of kinetic friction, ${\mu }_k$.
• The normal force acting on the object.

The coefficient of kinetic friction is a number that reflects how much friction a particular surface generates. For example, a higher value means more friction, while a lower value means less. In our exercise, ${\mu }_k=0.05$, indicating relatively low friction between the mug and the table.
To find the kinetic friction force, use the formula $$F_f=m{u}_{k}cdotN$$, where $F_f$ is the kinetic friction force, ${\mu }_k$ is the coefficient, and $N$ is the normal force.

Normal Force

The normal force is crucial in understanding how objects interact on surfaces. It acts perpendicular to the surface and supports the weight of an object resting on it. For our coffee mug, the normal force is equal to the weight of the mug since it is on a horizontal surface and there's no vertical acceleration. Thus, we use the formula $$N=m\cdot g$$, where $m$ is the mass and $g$ is the acceleration due to gravity, which is approximately 9.81 m/s².

• In our example, $N=0.12\cdot 9.81=1.1772\text{ N}$.
• This normal force serves as a basis for calculating kinetic friction.

The normal force can change if the surface is inclined or if additional forces act vertically on the object.

Net Force

Net force is the total force acting on an object after all individual forces are combined. Calculating net force is vital to determine how an object moves. For the coffee mug, the net force is the force you exert minus the kinetic friction force.
First, calculate the exerted force minus the friction force:

• Applied Force = 0.10 N
• Friction Force = 0.05886 N
• Net Force, $F_{\text{net}}=F-F_f=0.10-0.05886=0.04114\text{ N}$

The net force determines the mug's acceleration and subsequent motion across the table.
If more force were applied or if friction were lower, the net force would increase, making the mug move faster.

Work Formula

The work done on an object is the transfer of energy as a result of the force applied to move it. The amount of work depends on three main factors:

• The force applied ($F_{\text{net}}$.
• The distance the object moves in the direction of the force ($d$.
• The angle between the force applied and the direction of movement, noted as $\theta$.

The mathematical formula for work is $$W=F\cdot d\cdot cos(\theta )$$. In our scenario, since the force direction and the mug's motion direction are the same, the angle $\theta$ is 0, making $cos(\theta )=1$.
Plugging in the values from earlier:

• Net Force = 0.04114 N
• Distance = 0.15 m
• Work, $W=0.04114cdot0.15cdot1=0.006171\text{ J}$

This quantity of work signifies the energy transferred to push the mug across the table.