Problem 63 - College Physics

Question

The coefficients of static and kinetic friction between a \(50.0-\mathrm{kg}\) box and a horizontal surface are 0.500 and 0.400 respectively. (a) What is the acceleration of the object if a 250-N horizontal force is applied to the box? (b) What is the acceleration if the applied force is \(235 \mathrm{~N}\) ?

Step-by-Step Solution

Answer

(a) Acceleration is 1.08 m/s²; (b) Acceleration is 0 m/s².

Step 1: Calculate the Force of Static Friction

The first step is to find the force of static friction, which is the maximum force that must be overcome to start moving the box. The formula for the static frictional force is \( f_s = \mu_s \cdot N \), where \( \mu_s \) is the coefficient of static friction and \( N \) is the normal force. Since the box is on a horizontal surface, the normal force \( N \) equals the gravitational force on the box, \( m \cdot g \). Thus, \( N = 50.0 \cdot 9.8 = 490 \) N. Therefore, \( f_s = 0.500 \cdot 490 = 245 \) N.

Step 2: Determine Movement or No Movement for (a)

In part (a), the applied force is \( 250 \) N, which is greater than the static frictional force of \( 245 \) N calculated in Step 1. This indicates that the box will start moving as the applied force exceeds the force of static friction.

Step 3: Calculate Net Force and Acceleration (a)

Since the box moves, kinetic friction applies. The kinetic frictional force is \( f_k = \mu_k \cdot N = 0.400 \cdot 490 = 196 \) N. The net force acting on the box is then the applied force minus the kinetic frictional force: \( F_{net} = 250 - 196 = 54 \) N. The acceleration \( a \) can be found using Newton's second law \( F = m \cdot a \): \( a = \frac{F_{net}}{m} = \frac{54}{50.0} = 1.08 \) m/s².

Step 4: Determine Movement or No Movement for (b)

In part (b), the applied force is \( 235 \) N, which is less than the static frictional force of \( 245 \) N from Step 1. Since the applied force does not exceed the static frictional force, the box does not move, and thus the acceleration is \( 0 \) m/s².

Key Concepts

Static Friction

Static friction is the type of friction that prevents an object from moving when an external force is applied. It acts between two surfaces that aren't in relative motion. Static friction is a self-adjusting force, meaning it can change magnitude and direction depending on the applied force, up to a maximum point.
The formula for static friction is given by:

• \( f_s = \mu_s \cdot N \)

Where:

• \( f_s \) is the force of static friction
• \( \mu_s \) is the coefficient of static friction
• \( N \) is the normal force, which, for a box on a horizontal surface, equals the gravitational force \( m \cdot g \)

In the exercise, the static friction must be overcome by the applied force for the box to move. With a coefficient of static friction of 0.500 and a gravitational force of 490 N, the maximum static frictional force is 245 N.
If an applied force is equal to or less than this value, the box will stay at rest. Only when the applied force exceeds this static frictional force does the box start moving.

Kinetic Friction

Once an object starts moving, static friction is no longer relevant. This is when kinetic friction comes into play. Kinetic friction acts between surfaces in relative motion and is generally slightly less than static friction. It is often easier to keep an object in motion than to start moving it.
The formula for kinetic friction is:

• \( f_k = \mu_k \cdot N \)

Where:

• \( f_k \) is the force of kinetic friction
• \( \mu_k \) is the coefficient of kinetic friction
• \( N \) is the normal force

In our example, the kinetic friction occurs after the box begins to move, with a coefficient of kinetic friction of 0.400. This results in a kinetic frictional force of 196 N.
Kinetic friction provides resistance to the moving box, but it is lower than the resistance initially provided by static friction. The difference between the applied force and kinetic friction determines the box's net acceleration.

Newton's Second Law

Newton's second law of motion is pivotal in understanding how an object accelerates when forces act on it. It states that acceleration occurs when a net force acts on a mass, expressed mathematically as:

• \( F = m \cdot a \)

Where:

• \( F \) is the net force applied to the object
• \( m \) is the mass of the object
• \( a \) is the acceleration of the object

In the context of the exercise, we use this law to calculate the acceleration once the box begins moving. For instance, when a 250 N force is applied, and after accounting for kinetic friction, the net force on the box is 54 N.
By plugging these values into Newton's second law, we find the realistic acceleration of 1.08 m/s² for the box. Similarly, if the applied force doesn't exceed static friction, as with the 235 N force in part (b), there is no movement, making the acceleration 0 m/s².