Problem 42 - University Physics with Modern Physics

Question

A 6.50-kg instrument is hanging by a vertical wire inside a spaceship that is blasting off from rest at the earth's surface. This spaceship reaches an altitude of 276 m in 15.0 s with constant acceleration. (a) Draw a free-body diagram for the instrument during this time. Indicate which force is greater. (b) Find the force that the wire exerts on the instrument.

Step-by-Step Solution

Answer

(a) Tension force is greater. (b) Tension force is 79.6 N.

Step 1: Identify the Forces

Identify the forces acting on the instrument. They include the gravitational force, which pulls the instrument downwards, and the tension in the wire, which pulls it upwards. Because the spaceship is accelerating upwards, the tension must be greater than the gravitational force.

Step 2: Calculate Gravitational Force

The gravitational force (\( F_g \)) acting on the instrument can be calculated using the formula \( F_g = m \times g \), where \( m \) is the mass of the instrument (6.50 kg) and \( g \) is the acceleration due to gravity (9.8 \( m/s^2 \)). So, \( F_g = 6.50 \times 9.8 = 63.7 \text{ N} \).

Step 3: Determine Acceleration of the Spaceship

Use the kinematic equation \( s = ut + \frac{1}{2}at^2 \) to solve for the acceleration \( a \). With \( s = 276 \text{ m} \), \( u = 0 \), and \( t = 15 \text{ s} \), solve \( 276 = 0 + \frac{1}{2}a(15)^2 \). Rearranging gives: \( a = \frac{2 \times 276}{225} = 2.45 \text{ m/s}^2 \).

Step 4: Calculate the Apparent Weight

The tension in the wire (apparent weight) can be corrected using the formula\( W' = m(g + a) \). Substitute the values: \( m = 6.50 \text{ kg} \), \( g = 9.8 \text{ m/s}^2 \), \( a = 2.45 \text{ m/s}^2 \). Then \( W' = 6.50(9.8 + 2.45) = 6.50 \times 12.25 = 79.6 \text{ N} \).

Step 5: Conclude with the Tension Force

The force exerted by the wire is the apparent weight, which is 79.6 N. Since the wire tension is greater than the gravitational force, it confirms the acceleration is upward.

Key Concepts

Free-Body Diagram

Creating a free-body diagram is an essential step in analyzing the forces acting on an object. For this exercise, the diagram will represent the forces on a 6.50-kg instrument hanging in a spaceship. Begin by sketching the object, which in this instance is the instrument. Next, represent all forces with arrows that show their direction and magnitude.

• Gravitational Force: This is the force due to gravity, pulling the instrument downwards towards the Earth's center. It's represented as an arrow pointing downward, labeled with the formula, \( F_g = m \times g \).
• Tension Force: This is the force exerted by the wire, pulling the instrument upwards. The spaceship accelerates upward; thus, the tension force is greater than the gravitational force. It's denoted by an upward-pointing arrow.

Understanding the dominance of tension over gravitational force in this scenario helps explain why the instrument's apparent weight is greater than its true weight.

Apparent Weight

Apparent weight is the force felt by an object due to the net of all acting forces, in this case, experienced by the instrument inside the spaceship. Although the true weight of an object is due to gravity alone, apparent weight changes when other forces come into play, such as acceleration.

When the spaceship accelerates upward, the wire exerts an additional force on the instrument, perceived as an increased weight. The apparent weight can be calculated using the formula: \[W' = m(g + a) \]where \( m \) is the mass of the object, \( g \) is the gravitational acceleration (9.8 \text{ m/s}^2), and \( a \) is the spaceship's acceleration. In this problem, the apparent weight becomes greater than the weight due to gravity alone because the upward acceleration adds to the gravitational pull. This is why passengers feel heavier when an elevator starts moving up sharply.

Gravitational Force

The gravitational force is a fundamental concept described by Newton's Laws of Motion and is essential for understanding motion within Earth's gravitational field. This force pulls objects towards the center of the Earth and can be calculated using the equation:\[ F_g = m \times g \] where \( m \) represents the mass of the object and \( g \) is the standard acceleration due to gravity, typically taken as 9.8 \text{ m/s}^2 near the Earth's surface.

For the instrument in the spaceship, the gravitational force acting downward is calculated to be 63.7 N. This force is constant and always acts towards the center of the Earth. However, when additional forces, such as the spaceship's acceleration, come into play, the apparent weight becomes more noticeable than the gravitational force alone. Understanding gravitational force helps in decoding many natural phenomena, like why objects fall or why they feel heavier when we accelerate upwards.