Question

You slide a 0.12-kg coffee mug 0.15 m across a table. The force you exert is horizontal and of magnitude 0.10 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

0.00618 J of work is done on the mug.

Step-by-step solution

Identify Forces and Parameters

To solve for the work done on the coffee mug, first identify the forces acting on it. We have a horizontal applied force of 0.10 N, the mass of the mug is 0.12 kg, the displacement is 0.15 m, and the coefficient of kinetic friction between the mug and the table is 0.05.

Calculate Normal Force

The normal force () is equal to the gravitational force acting on the mug, which is given by the formula: \( F_{n} = m \cdot g \). Substituting the values, we have:\[ F_{n} = 0.12 \, \text{kg} \times 9.8 \, \text{m/s}^2 = 1.176 \, \text{N} \]

Calculate Frictional Force

The frictional force () is calculated using the formula: \( F_{f} = \mu \, \cdot F_{n} \), where \( \mu \) is the coefficient of kinetic friction. Substituting the given values, we have:\[ F_{f} = 0.05 \times 1.176 \, \text{N} = 0.0588 \, \text{N} \]

Net Force Calculation

The net force () acting on the mug is the applied force minus the frictional force:\[ F_{ ext{net}} = F_{ ext{applied}} - F_{f} = 0.10 \, \text{N} - 0.0588 \, \text{N} = 0.0412 \, \text{N} \]

Calculate Work Done

The work done () on the mug is given by the formula: \( W = F_{ ext{net}} \cdot d \), where \( d \) is the displacement. Substituting the calculated net force and displacement, we have:\[ W = 0.0412 \, \text{N} \times 0.15 \, \text{m} = 0.00618 \, \text{J} \]

Conclusion

Finally, the work done on the coffee mug by the net force is 0.00618 Joules.

Key concepts

Kinetic Friction

Kinetic friction is the force that opposes the motion of two surfaces sliding past each other. It is crucial in determining how much work needs to be done to move an object. Whenever surfaces are in motion relative to each other, kinetic friction comes into play. This force depends on two main factors: the normal force and the coefficient of kinetic friction.

In our example with the coffee mug, the kinetic frictional force is calculated using the formula:

• \( F_{f} = \mu \cdot F_{n} \)

where \( \mu \) is the coefficient of kinetic friction (given as 0.05) and \( F_{n} \) is the normal force. The result, 0.0588 N, represents the force that resists the mug's sliding motion. Understanding kinetic friction helps us understand why it's more challenging to move objects across certain surfaces than others. It also explains how much energy is needed to maintain motion.

Normal Force

The normal force is an essential concept in physics, particularly when analyzing forces on a surface. It is the perpendicular force exerted by a surface to support the weight of an object resting on it. For objects laid flat on a horizontal surface, like our coffee mug, the normal force balances the gravitational force. This balance prevents objects from accelerating into the surface.
The normal force can be calculated using:

• \( F_{n} = m \times g \)

where \( m \) is the object's mass, and \( g \) is the acceleration due to gravity, approximately \( 9.8 \, \text{m/s}^2 \). In our scenario, with a mug of mass 0.12 kg, the normal force is 1.176 N. This force plays a crucial role in determining the frictional force and hence the overall work needed to move the mug across the table.

Net Force

The net force is the total force acting on an object. It is the vector sum of all the individual forces applied to that object. Understanding net force helps us determine the motion of an object, like whether it will accelerate, decelerate, or move at a constant speed.

In the case of the sliding coffee mug, the net force is the result of subtracting the frictional force from the applied force:

• \( F_{\text{net}} = F_{\text{applied}} - F_{f} = 0.10 \, \text{N} - 0.0588 \, \text{N} \)

This calculated net force of 0.0412 N is what allows the mug to continue moving over the distance of 0.15 m. Without this net force, the mug would eventually come to a stop due to friction. The concept of net force is fundamental in understanding the dynamics of motion and energy transfer.

Displacement

Displacement refers to the change in position of an object. It is a vector quantity, meaning it has both magnitude and direction. In physics, displacement is crucial for calculating work done on an object, which is the energy transferred when a force moves an object over a distance.
In the case of our coffee mug, the displacement is 0.15 m. This is the measure of how far the mug moved across the table. When calculating work, we use this distance alongside the net force:

• \( W = F_{\text{net}} \cdot d \)

The work done, calculated as 0.00618 Joules, indicates how much energy was transferred to move the mug across the table. Displacement plays a vital role because the amount of work done is directly proportional to the distance moved.

Gravitational Force

Gravitational force is a fundamental force in physics, responsible for attracting objects with mass towards each other. On Earth, it gives weight to objects, pulling them toward the center of the planet. This force is what holds the coffee mug on the table. Without it, objects would float away.
The gravitational force acting on any object with mass can be calculated using:

• \( F_{g} = m \times g \)

where \( m \) is the mass and \( g \) is the gravitational acceleration (~9.8 m/s² on Earth). For our coffee mug, this force equals 1.176 N, which is exactly balanced by the normal force. Gravitational force not only influences the normal force but also plays a pivotal role in defining other force interactions and energy considerations.