Question

A 53-lb \((=240-\mathrm{N})\) trunk rests on the floor. The coefficient of static friction between the trunk and the floor is \(0.41\), while the coefficient of kinetic friction is \(0.32 .(a)\) What is the minimum horizontal force with which a person must push on the trunk to start it moving? (b) Once moving, what horizontal force must the person apply to keep the trunk moving with constant velocity? (c) If, instead, the person continued to push with the force used to start the motion, what would be the acceleration of the trunk?

Short answer

The minimum force needed to start moving the trunk is 98.4N, and the force required to keep it moving at a constant velocity is 76.8N. If the initial force is continually applied, the trunk will accelerate at 0.88 m/s^2.

Step-by-step solution

- Calculate force required to overcome static friction

To find the minimum force required to start the trunk moving, you first find the static friction, \(f_{s}\), which is given by the equation \(f_{s} = \mu_{s} * N\), where \(\mu_{s}\) is the coefficient of static friction and \(N\) is the normal force. Because the trunk is on a flat floor, the normal force is simply equal to the weight of the trunk. Multiply the static friction coefficient (\(.41\)) by the weight of the trunk ( \(240N\) ), to obtain \(f_{s} = .41 * 240 = 98.4 N\). This is the static friction holding the trunk in place.

- Calculate the force required to overcome kinetic friction

Once moving, the trunk encounters kinetic friction. The frictional force of kinetic friction, \(f_{k}\), is calculated by multiplying the coefficient of kinetic friction (\(.32\)) by the normal force ( trunk's weight, \(240N\)). This gives us \(f_{k} = .32 * 240 = 76.8 N\). This is the horizontal force required to keep the trunk moving at constant velocity.

- Calculate the acceleration

If the person continues to push with the initial force used to overcome static friction ( \(98.4N\)), subtract the force of kinetic friction ( \(76.8N\)) from the applied force ( \(98.4N\)). This gives an unbalanced force of \(21.6N\). Apply Newton's second law of motion, \(F=ma\), to find the acceleration, \(a\). The mass, \(m\), of the trunk is the weight divided by the acceleration due to gravity (\(9.8 m/s^2\)): \(m = 240N / 9.8 m/s^2 = 24.49 kg\). Insert this into the equation with the unbalanced force, to find \(a = F/m = 21.6N / 24.49 kg = 0.88 m/s^2\).

Key concepts

Static Friction

Static friction is the force that keeps an object at rest when a force is applied until the force reaches a certain threshold. Imagine trying to push a heavy trunk across the floor. When you initially push it, it doesn't move instantly because of static friction.
This force is determined by two main factors:

• The coefficient of static friction (\(\mu_s\): a measure of how grippy the surfaces in contact are)
• The normal force (\(N\): the force perpendicular to the surfaces in contact, usually the weight of the object if it's on a horizontal floor)

Static friction can be calculated using the formula:\[f_{s} = \mu_{s} \times N\]The friction force will keep an object stationary until the applied force exceeds \(f_s\). In our exercise, we find the static friction force using \(\mu_s = 0.41\) and \(N = 240 \text{ N}\), leading to a static friction of \(98.4 \text{ N}\). This is the force you need to overcome to start the trunk moving.

Kinetic Friction

Once the trunk starts moving, static friction gives way to kinetic friction, which is slightly lower. Kinetic friction is the force that acts against the motion of two surfaces sliding past each other.
Kinetic friction is less than static friction because once an object is in motion, it's easier to keep it moving than to start moving. The calculation of kinetic friction involves:

• The coefficient of kinetic friction (\(\mu_k\)): which is usually less than the static coefficient
• The normal force (\(N\))

The formula for kinetic friction is similar to static friction:\[f_{k} = \mu_{k} \times N\]In the exercise, \(\mu_k = 0.32\) and the normal force remains \(N = 240 \text{ N}\). Thus, the kinetic friction is \(76.8 \text{ N}\). This means once the trunk is in motion, you'll need to apply \(76.8 \text{ N}\) horizontally to keep it moving at a steady speed.

Newton's Second Law

Newton's Second Law is a fundamental principle in physics that links force, mass, and acceleration. This law is expressed with the formula:\[F = ma\]where \(F\) is the net force applied, \(m\) is the mass of the object, and \(a\) is the acceleration produced. This allows us to predict how an object will move when acted on by a force.
Upon applying this law in our exercise, after overcoming static friction (with a force of \(98.4 \text{ N}\)), if we maintain this force, we subtract the kinetic friction force \(76.8 \text{ N}\) to find the net force. This leaves us with a net force of \(21.6 \text{ N}\).
To find acceleration, divide this net force by the trunk's mass \(24.49 \text{ kg}\):\[a = \frac{21.6}{24.49} = 0.88 \text{ m/s}^2\]So, the trunk will accelerate at \(0.88 \text{ m/s}^2\) if the initial pushing force is maintained after overcoming static friction.