Problem 23 - College Physics

Question

\(\bullet$$\bullet\) A large fish hangs from a spring balance supported from the roof of an elevator. (a) If the elevator has an upward accel- eration of 2.45 \(\mathrm{m} / \mathrm{s}^{2}\) and the balance reads \(60.0 \mathrm{N},\) what is the true weight of the fish? (b) Under what circumstances will the balance read 35.0 \(\mathrm{N} ?\) (c) What will the balance read if the ele- vator cable breaks?

Step-by-Step Solution

Answer

(a) True weight is approximately 42.0 N. (b) Balance reads 35.0 N with downward accel of ~3.5 m/s². (c) Balance reads 0 N if cable breaks.

Step 1: Analyze Forces in the Accelerating Elevator

In an elevator with upward acceleration, the apparent weight is increased by the force due to the elevator's acceleration. This means we consider both the gravitational force and the additional force because of acceleration.

Step 2: Apply Newton's Second Law

The net force on the fish can be described by the formula: \[ F_{net} = m imes a = T - m imes g \] where \( T \) is the tension (or reading of the scale), \( m \) is the mass of the fish, \( a \) is the elevator's acceleration, and \( g \) is the acceleration due to gravity (\( 9.8 \, \text{m/s}^2 \)).

Step 3: Calculate Mass of the Fish

From the formula in Step 2, solve for mass \( m \): \[ T = m(g + a) \quad \Rightarrow \quad m = \frac{T}{g + a} = \frac{60.0}{9.8 + 2.45} \text{ kg} \] Calculate \( m \) using the given values, \( T = 60.0 \text{ N} \), \( a = 2.45 \text{ m/s}^2 \).

Step 4: Calculate True Weight of the Fish

The true weight of the fish \( W \) is given by \( W = m imes g \). Calculate \( W \) using the mass \( m \) found in Step 3 and \( g = 9.8 \, \text{m/s}^2 \).

Step 5: Determine Conditions for Balance Reading 35.0 N

Using \( T = m(g + a) \), set \( T \) to 35.0 N and solve for \( a \): \[ 35 = m(g + a) \quad \Rightarrow \quad a = \frac{35}{m} - g \] Substitute \( m \) from Step 3 to find the required acceleration.

Step 6: Calculate Balance Reading when Cable Breaks

If the elevator cable breaks, the acceleration is equal to \( g \) (free fall conditions), meaning the net force is zero and the balance reads 0 N because both the fish and the balance are accelerating downwards at the same rate.

Key Concepts

apparent weight

Apparent weight describes how heavy or light an object seems due to external forces, such as acceleration or deceleration. In physics, apparent weight is different from actual weight, which is the force due to gravity. When in motion, such as inside an accelerating elevator, an object's apparent weight can change. If an elevator accelerates upwards, an object seems heavier because the force required to counteract gravity increases. This happens because: - The forces acting on the object include both gravity ( g ) and the elevator’s upward acceleration ( a ). - The total force is m(g + a) , where m is the object’s mass. This combined force affects the balance readings. For example, if a spring balance reads 60 N while accelerating upwards, it is because this extra force increases the apparent weight.

elevator physics problem

An elevator physics problem often involves Newton's Second Law, focusing on forces acting on an object inside an elevator. These problems help illustrate principles like apparent weight and forces in motion.To solve such problems, one must analyze the changes in forces when the elevator:- Moves upward: Increases the apparent weight due to added force from acceleration.- Moves downward: Decreases the apparent weight if it decelerates or free falls.Consider a scenario where a fish is suspended in an elevator with an upward acceleration of 2.45 m/s². Using Newton's second law: \[ F_{net} = m \times a = T - m \times g \]To find the actual mass or weight:- Rearrange to solve for mass \( m \): \[ m = \frac{T}{g + a} \]- Calculate the true weight with \( W = m \times g \).This methodology helps determine mysteries like why a balance reads different numbers based on how the elevator moves.

free fall conditions

Free fall occurs when the only force acting on an object is gravity, meaning that any resistance, like air friction, is negligible. When an elevator cable breaks, both the elevator and its contents fall freely, accelerating downward at gravity's rate.In this state:- The object, such as a fish on a spring balance, no longer exerts force on the balance.- Hence, the apparent weight becomes zero because the support force from the balance is gone.Using Newton's second law, the scenario implies:- \( F_{net} = 0 \) because both the object and the balance fall together.When conceptualizing such problems:- It's useful to remember that in true free fall, all objects experience the same acceleration: \( g = 9.8 \, \text{m/s}^2 \).- With no upward or downward force exerted by the object onto the balance, scales read zero, confirming the free fall conditions.