Problem 32 - College Physics

Question

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.

Step-by-Step Solution

Answer

Net force is zero when the train has uniform velocity. It's not zero when the train is accelerating.

Step 1: Understanding the Scenario

In this problem, we are dealing with a ball hanging inside a moving train. In both cases provided, the ball is observed from the train's reference frame.

Step 2: Case (a): Uniform Velocity

When the train is moving with a uniform velocity, it means there is no acceleration and the forces on the ball are balanced. The free-body diagram will show two forces: gravity acting downward and tension from the string acting upward. The tension force equals the weight of the ball, so the net force is zero.

Step 3: Free-Body Diagram for Uniform Velocity

Draw the free-body diagram: draw a dot for the ball. Draw vector arrows from the center of the ball. Point the arrow of the tension force (T) upwards and the gravity force (mg) downwards. Both arrows should be equal in length to reflect balanced forces.

Step 4: Net Force in Uniform Velocity Case

With no acceleration in the horizontal direction and balanced forces vertically, the net force on the ball is zero.

Step 5: Case (b): Speeding Up Uniformly

If the train is speeding up uniformly, this means there is horizontal acceleration to the east. In the reference frame of the train, the ball's free-body diagram will also show a force acting horizontally to the west, in addition to the vertical tension and gravitational forces.

Step 6: Free-Body Diagram for Speeding Up

Draw the free-body diagram: draw a dot for the ball. Draw vector arrows from the center of the ball. The tension (T) now inclines slightly backward, the gravity force (mg) still points downward, and a fictitious force equivalent to the acceleration acts horizontally to the west.

Step 7: Net Force in Speeding Up Case

Though vertically balanced by tension and gravity, horizontally, the ball experiences a net force due to the train's acceleration. Thus, the net force is not zero because of this horizontal component.

Key Concepts

Uniform Velocity

When discussing uniform velocity, we are dealing with an object moving at a constant speed in a straight line. In the context of a free-body diagram, this is important because it indicates that there is no net acceleration acting on the ball.

In such a scenario, the key point is that the forces acting on the object must be balanced. This means that the force of gravity, which pulls the ball downwards, is exactly counteracted by the tension in the string pulling it upwards. Thus, the free-body diagram in this case will show:

• A downward arrow labeled \( mg \) representing the gravitational force.
• An upward arrow labeled \( T \) representing the tension in the string.

Given that these forces are balanced, the net force on the ball is zero. The ball remains stationary relative to the train car, so there is no horizontal component to consider in this simplistic scenario.

Net Force

The net force is defined as the vector sum of all the forces acting on an object. It dictates the change in motion of the object according to Newton's second law of motion. In the scenario where the train moves with a uniform velocity, the net force is zero because the forces are balanced vertically with no horizontal forces acting on the ball.

However, in the case where the train is accelerating, the net force is no longer zero. The ball experiences an additional horizontal force due to the train's acceleration. This fictitious force, often referred to in the object's frame of reference, acts opposite to the direction of the train’s acceleration.

Therefore, a non-zero net force provides the acceleration component needed to explain the ball’s change in motion within the reference frame of the accelerating train.

Tension and Gravity

Tension and gravity are the two main forces to consider when analyzing the ball hanging from a string in the moving train. Gravity, acting downward, is the force due to the Earth pulling the ball towards its center. It can be expressed as \( mg \), where \( m \) is the mass of the ball and \( g \) is the acceleration due to gravity (approximately \( 9.8 \, m/s^2 \)).

Tension, on the other hand, is the force exerted through the string to keep the ball suspended. It acts upward to oppose gravity. In a uniform velocity scenario, the tension force perfectly matches the gravitational force, meaning \( T = mg \). This balance is crucial for maintaining a state of equilibrium.

In the case of horizontal acceleration, tension also adjusts to include an angle that accounts for both the gravitational component and the fictitious force arising from the train’s acceleration.

This results in the tension vector inclining slightly towards the direction of the fictitious force.

Horizontal Acceleration

In physics, horizontal acceleration is the change in velocity in a horizontal direction. When the train is speeding up, it introduces acceleration in the horizontal plane, affecting the forces experienced by objects within the train.

For the ball hanging inside the train, this added acceleration causes a shift in how forces are balanced. The train’s eastward acceleration results in the appearance of a fictitious force that acts in the opposite direction, from the perspective of an observer inside the train.

The free-body diagram for this accelerating scenario must accommodate this additional force:

• Gravity (\( mg \)), acting directly downward.
• Tension (\( T \)), slightly inclined from the vertical line due to the added fictitious force.
• The fictitious force, acting horizontally to the west. This force doesn’t alter the total magnitude of tension but changes its direction slightly.

This rearrangement of forces reflects a net force acting on the ball, which is no longer zero due to this horizontal component.