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Chapter 4: Problem 29

A ball is hanging from a long string that is tied to the ceiling of a train car traveling eastward on horizontal tracks. An observer inside the train car sees the ball hang motionless. Draw a clearly labeled free-body diagram for the ball if (a) the train has a uniform velocity and (b) the train is speeding up uniformly. Is the net force on the ball zero in either case? Explain.

Short Answer

Expert verified
In case (a), where the train is moving at a uniform velocity, the net force on the ball is zero. However, in case (b), where the train is accelerating, there is a net force acting on the ball towards the direction of the train's acceleration.

Step by step solution

01

Situation (a): Train moving at uniform velocity

When the train is moving at a uniform velocity, it means there is no acceleration in any direction. Thus from the observer's point of view, the ball appears to be at rest. Besides, according to Newton's first law, an object in motion continues in motion with the same speed and in the same direction unless acted upon by an unbalanced force. Thus, even though the train car and hence the string provides an eastward force (cause the train is moving eastward), there's no change in the velocity of the ball, so the net force on it is null. So the free-body diagram for the ball would show only two forces: the downward gravitational force \(F_g\), and the upward tension force \(T\) from the string. These forces cancel out, so the net force on the ball is zero.
02

Situation (b): Train is speeding up

If the train is accelerating uniformly eastward, a force is acting on the train and everything inside it towards the east. For an observer in the train, the ball appears to incline towards the west. The ball's free-body diagram now has three forces: the gravitational force \(F_g\), the tension \(T\) in the string, and a force due to the train鈥檚 acceleration \(a\) towards the east, \(F = m \cdot a\). These forces are not balanced, which means there's a net force acting on the ball.
03

Analyzing the net force in each case

In case (a), the net force on the ball is zero because the forces acting on it are balanced. However, in case (b), there is a net force acting on the ball due to the train鈥檚 acceleration. Newton's second law states that the net force on an object is equal to the mass of the object multiplied by the acceleration of the object. Thus, the net force on the ball would be the vector sum of three forces \(F_g\), \(T\) and \(F_{acceleration}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Free-Body Diagrams
A free-body diagram is a visual tool used in physics to represent all the forces acting on an object. It helps us analyze the situation by breaking down complex interactions into simpler components. When creating a free-body diagram for an object, like the ball in the train car, we identify each force acting on it.
For example, in situation (a), where the train moves with uniform velocity, the free-body diagram for the ball includes:
  • the gravitational force downward (\( F_g \)),
  • the tension in the string upward (\( T \)).
These forces balance each other out, leading to a total net force of zero. When the forces are balanced, the object does not accelerate, maintaining its state of motion.
In situation (b), however, as the train speeds up, an additional force acts on the ball due to the acceleration, resulting in:
  • the gravitational force (\( F_g \)),
  • the tension (\( T \)),
  • a force from the train's acceleration (\( F = m \cdot a \)).
This results in an unbalanced net force, causing the ball to accelerate.
Concept of Uniform Velocity
Uniform velocity occurs when an object travels in a straight line at a constant speed. In physics, this means there is no net force acting on the object, so there is no acceleration.
In the context of the exercise, when the train has uniform velocity, the ball hanging inside remains motionless relative to the observer on the train. Although the train is moving, the ball does not change its state of motion because the forces acting on it are balanced.
Newton's first law, often referred to as the law of inertia, supports this by stating that an object at rest will stay at rest, and an object in motion will continue in motion at a constant velocity unless acted upon by a net external force. Thus, in the train's constant velocity situation, the ball does not experience acceleration, emphasizing the balance between the downward gravitational force and the upward tension force.
Acceleration and Its Effects
Acceleration is a measure of how quickly an object changes its velocity. It occurs when a net force is applied, resulting in changes in speed or direction.
In scenario (b) from the exercise, the train's acceleration changes the state of motion for objects inside it, such as the ball. When the train accelerates eastward, a force due to this acceleration acts on the ball, observed as a westward inclination by someone in the train. This new force is calculated by the formula \( F = m \cdot a \), where \( m \) is the mass of the ball and \( a \) is the train's acceleration.
Newton's second law explains this concept well, stating that the net force acting on an object is equal to the mass of the object multiplied by its acceleration. In this situation, the net force is not zero, causing the ball to accelerate westward in response to the train's eastward acceleration, revealing the dynamic nature of forces in motion.

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Most popular questions from this chapter

A box rests on a frozen pond, which serves as a frictionless horizontal surface. If a fisherman applies a horizontal force with magnitude \(48.0 \mathrm{~N}\) to the box and produces an acceleration of magnitude \(2.20 \mathrm{~m} / \mathrm{s}^{2}\), what is the mass of the box?

Starting at time \(t=0\), net force \(F_{1}\) is applied to an object that is initially at rest. (a) If the force remains constant with magnitude \(F_{1}\) while the object moves a distance \(d\), the final speed of the object is \(v_{1} .\) What is the final speed \(v_{2}\) (in terms of \(v_{1}\) ) if the net force is \(F_{2}=2 F_{1}\) and the object moves the same distance \(d\) while the force is being applied? (b) If the force \(F_{1}\) remains constant while it is applied for a time \(T,\) the final speed of the object is \(v_{1} .\) What is the final speed \(v_{2}\) (in terms of \(v_{1}\) ) if the applied force is \(F_{2}=2 F_{1}\) and is constant while it is applied for the same time \(T ?\) In a later chapter we'll call force times distance work and force times time impulse and associate work and impulse with the change in speed.)

Two crates, one with mass \(4.00 \mathrm{~kg}\) and the other with mass \(6.00 \mathrm{~kg},\) sit on the frictionless surface of a frozen pond, connected by a light rope (Fig. P4.37). A woman wearing golf shoes (for traction) pulls horizontally on the \(6.00 \mathrm{~kg}\) crate with a force \(F\) that gives the crate an acceleration of \(2.50 \mathrm{~m} / \mathrm{s}^{2}\). (a) What is the acceleration of the \(4.00 \mathrm{~kg}\) crate? (b) Draw a free-body diagram for the \(4.00 \mathrm{~kg}\) crate. Use that diagram and Newton's second law to find the tension \(T\) in the rope that connects the two crates. (c) Draw a free-body diagram for the \(6.00 \mathrm{~kg}\) crate. What is the direction of the net force on the \(6.00 \mathrm{~kg}\) crate? Which is larger in magnitude, \(T\) or \(F ?\) (d) Use part (c) and Newton's second law to calculate the magnitude of \(F\).

Estimate the average force that a major-league pitcher exerts on the baseball when he throws a fastball. Express your answer in pounds. In your solution, list the quantities for which you estimate values and any assumptions you make. Do a web search to help determine the values you use in making your estimates.

Boxes \(A\) and \(B\) are in contact on a horizontal, frictionless surface (Fig. E4.23). Box A has mass \(20.0 \mathrm{~kg}\) and box \(B\) has mass \(5.0 \mathrm{~kg} .\) A horizontal force of \(250 \mathrm{~N}\) is exerted on box \(A\). What is the magnitude of the force that box \(A\) exerts on box \(B ?\)
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