Question

Which of the following statements is true of movement on a plane with friction? Acceleration is a function of applied force only. More force is needed to accelerate a stationary object than an identical moving object. The force of friction is independent of the mass of objects. (A) I only (B) II only (C) I and II only (D) I and III only

Short answer

The correct answer is (B) II only.

Step-by-step solution

- Analyze Statement I

Statement I reads, 'Acceleration is a function of applied force only.' Using Newton's Second Law of Motion, we know that \( F = ma \), where \( F \) is the force applied, \( m \) is the mass, and \( a \) is the acceleration. However, this is true only in a frictionless context. When friction is present, the frictional force must also be considered. Therefore, acceleration is not only a function of applied force, making Statement I false on a plane with friction.

- Analyze Statement II

Statement II reads, 'More force is needed to accelerate a stationary object than an identical moving object.' This is true because of static friction, which needs to be overcome to start moving the object. Once the object is in motion, kinetic friction, which is usually lower, takes over. Therefore, more force is indeed needed to accelerate a stationary object.

- Analyze Statement III

Statement III reads, 'The force of friction is independent of the mass of objects.' This statement is false because frictional force is given by \( f = \text{μ}N \), where \( \text{μ} \) is the coefficient of friction and \( N \) is the normal force. The normal force is directly related to the mass of the object, making friction dependent on mass.

- Determine the Correct Answer

Based on the analysis, Statement I is false, Statement II is true, and Statement III is false. Therefore, the correct answer is (B) II only.

Key concepts

frictional force

Frictional force is a resistive force that opposes the motion of an object. It acts parallel to the surface with which the object is in contact.
This force is responsible for making it harder to push or pull objects across surfaces. The frictional force can be calculated using the formula:
\[ f = \text{μ}N \]
where \( f \) is the frictional force, \( \text{μ} \) is the coefficient of friction, and \( N \) is the normal force.
The normal force is typically equal to the weight of the object if the surface is horizontal. Frictional force comes in two main flavors: static friction and kinetic friction, each playing its role in different contexts.

static friction

Static friction is the force that keeps an object at rest when a force is applied, preventing it from moving. This type of friction is always equal and opposite to the applied force until the object starts to move.
When you want to move a stationary object, you need to apply a force that exceeds the maximum static friction. The equation for static friction is:
\[ f_s \, \text{(max)} = μ_s N \]
Here, \( f_s \, (\text{max}) \) is the maximum static friction, \( μ_s \) is the coefficient of static friction, and \( N \) is the normal force. Static friction adjusts itself to exactly balance the applied force until it reaches its maximum value, at which point the object begins to slide.

kinetic friction

Kinetic friction occurs when an object is already moving. Unlike static friction, kinetic friction has a constant value. This value is usually lower than maximum static friction, which is why it takes less force to keep an object moving than to start it moving.
The equation for kinetic friction is:
\[ f_k = μ_k N \]
Here, \( f_k \) is the kinetic friction, \( μ_k \) is the coefficient of kinetic friction, and \( N \) is the normal force. Knowing the difference between static and kinetic friction helps us understand why it's harder to start moving an object than to keep it moving.

acceleration

Acceleration is the rate at which an object's velocity changes over time. According to Newton's Second Law of Motion, acceleration is produced when a force acts on a mass.
The law is often expressed as:
\[ F = ma \]
Here, \( F \) is the net force applied to an object, \( m \) is the object's mass, and \( a \) is the acceleration. However, when frictional forces are present, we must modify this equation to:
\[ F_{\text{net}} = ma \]
where \( F_{\text{net}} \) takes both the applied force and the frictional force into account. So in a scenario involving friction, acceleration is influenced by both the applied force and the frictional force working against it.