Problem 80 - University Physics with Modern Physics

Question

A hammer is hanging by a light rope from the ceiling of a bus. The ceiling of the bus is parallel to the roadway. The bus is traveling in a straight line on a horizontal street. You observe that the hammer hangs at rest with respect to the bus when the angle between the rope and the ceiling of the bus is \(67^{\circ} .\) What is the acceleration of the bus?

Step-by-Step Solution

Answer

The acceleration of the bus is approximately \(23.1 \, \text{m/s}^2\).

Step 1: Understanding the Forces Involved

When the hammer is hanging at an angle of \(67^{\circ}\) with the ceiling, it suggests that there's a force component causing this hanging angle. The forces acting on the hammer are gravity (acting downward) and tension in the rope (directed along it). The horizontal component of the tension force must be providing the centripetal force needed for the bus's acceleration because the hammer is not moving sideways.

Step 2: Analyze the Angle of the Rope

The angle \(\theta = 67^{\circ}\) formed by the rope with the vertical is crucial because it allows us to resolve the tension into components. The vertical component of tension balances the weight of the hammer (gravity), and the horizontal component provides the horizontal acceleration due to the bus's motion.

Step 3: Set Up the Equations

Using Newton's second law, we can resolve the tension \(T\) into components: \( T \cos(\theta) = mg \) (vertical balance) and \( T \sin(\theta) = ma \) (horizontal acceleration), where \(m\) is the mass of the hammer, \(g\) is the acceleration due to gravity (9.8 m/s²), and \(a\) is the acceleration of the bus.

Step 4: Solve for Acceleration

From the two equations, we eliminate \(T\) by dividing \( T \sin(\theta) = ma \) by \( T \cos(\theta) = mg \), obtaining \( \tan(\theta) = \frac{a}{g} \). Solving for \(a\) gives \( a = g \tan(\theta) \). Substitute \(g = 9.8 \, \text{m/s}^2\) and \(\theta = 67^{\circ}\) (convert degrees to radians if necessary) into this equation.

Step 5: Calculation

Calculate \( \tan(67^{\circ}) \) and multiply by \(9.8\), it comes to approximately \( 9.8 \times 2.355 \approx 23.1 \, \text{m/s}^2 \). Therefore, the acceleration of the bus is \(23.1 \, \text{m/s}^2\).

Key Concepts

Centripetal Force

When the hammer inside the bus hangs at an angle, it indicates that there is a force acting towards the center of the circular path the hammer would take if the bus continued moving in a curved line. This force is known as the centripetal force. It is not a separate force but rather a result of the components of other forces acting together to keep the hammer at rest relative to the bus. - The tension in the rope provides the centripetal force needed, preventing sideways motion of the hammer. - Since the bus is accelerating, this centripetal force is essentially the horizontal component of the tension force. By understanding that centripetal force is the force that keeps an object moving in a curved path, we can see how it relates to other forces involved—such as tension and gravity—when resolving them into components.

Components of Forces

Forces acting on the hammer can be broken down into components to make complex problems easier to solve. When dealing with angled forces, like tension in a rope, it's helpful to use trigonometric functions to separate the force into its vertical and horizontal components. - The vertical component of the tension force counteracts the gravitational force pulling the hammer downwards. This means the hammer stays at a constant height without falling. - The horizontal component of the tension is responsible for the sideways or horizontal forces that we term as centripetal force in this scenario. By resolving forces into components, it becomes simpler to apply Newton's Second Law to different directions independently, which aids in understanding the motion and equilibrium of the system.

Tension in a Rope

The rope's tension is a crucial force in this situation, providing the necessary support and balance for the hammer. When a rope has tension, it means that the rope is being pulled tight between two forces, maintaining equilibrium. - Tension acts along the length of the rope, trying to pull equally on objects at both ends. - In this scenario, the rope's tension balances the downward pull of gravity through its vertical component. Meanwhile, the horizontal component of tension supplies the centripetal force needed for acceleration. This dual role of tension in both horizontal and vertical planes helps illustrate why understanding and calculating tension accurately is important in dynamics and system analysis.

Angle of Inclination

The angle of inclination is an important factor because it affects how forces are resolved into vertical and horizontal components. In this exercise, the angle between the rope and the vertical is specified as 67 degrees. - This angle plays a critical role because it determines the ratio of the horizontal to vertical force components. - Trigonometric functions like sine and cosine are used with the angle to calculate these components. - Through calculations, the angle helps us find how much of the tension contributes to balancing gravity versus contributing to the bus's acceleration. Understanding how angles affect force distributions can aid in solving a variety of problems involving motion and equilibrium, reinforcing the interconnectedness of geometry and physics.