Problem 24 - College Physics

Question

A rocket is far away from all planets and stars, so gravity is not a consideration. It is using its rocket engines to accelerate upward with an acceleration \(a=9.80 \mathrm{~m} / \mathrm{s}^{2} .\) On the floor of the main deck is a crate (object with brick pattern) with a mass of \(75.0 \mathrm{~kg}\) (a) How many forces are acting on the crate: (v Fig. 4.35). (1) zero; (2) one; (3) two; (4) three? (b) Determine the normal force on the crate and compare it to the normal force the crate would experience if it were at rest on the surface of the Earth.

Step-by-Step Solution

Answer

(a) Two forces act on the crate; (b) The normal force is 735 N, equal to its weight on Earth.

Step 1: Identify Forces Acting on the Crate

In space, where gravity is not a factor, the only force acting on the crate initially is the force from the rocket's acceleration acting upwards. However, recall that any object on a surface experiences a normal force from that surface. Thus, the two forces on the crate are the normal force and the force due to the rocket's acceleration. Hence, there are two forces acting on the crate.

Step 2: Calculate the Force due to Rocket's Acceleration

The force due to the rocket's acceleration is calculated using Newton's second law, \(F = ma\). Here, the mass \(m = 75.0 \text{ kg}\) and the acceleration \(a = 9.80 \text{ m/s}^2\). Thus, the force \(F = 75.0 \times 9.80 = 735 \text{ N}\).

Step 3: Determine the Normal Force on the Crate

In the non-gravitational field, the normal force \(N\) balances the acceleration force to ensure the crate does not move vertically relative to the rocket. Thus, the normal force is equal in magnitude to the acceleration force acting on the crate. Therefore, the normal force \(N = 735 \text{ N}\).

Step 4: Compare Normal Force to Earth's Surface Condition

On Earth's surface, the normal force acting on a stationary crate would be its weight, calculated as \(mg\), where \(g = 9.80 \text{ m/s}^2\) is the gravitational acceleration. The weight \(W = 75.0 \times 9.80 = 735 \text{ N}\). Thus, the normal force on the crate is the same as it would experience on Earth's surface.

Key Concepts

Normal Force

Understanding the normal force is crucial in physics as it often acts as a counterbalance to other forces. In simple terms, the normal force is the force exerted by a surface to support the weight of an object resting on it. It acts perpendicular to the surface.
In the context of the problem, even though the crate is in space where gravity isn't a concern, the rocket's floor still exerts a normal force. This is because the crate is being pushed against the rocket's floor due to the rocket's upward acceleration. Think of it like being squashed into your seat when a car accelerates quickly.

• Normal force doesn't necessarily equal the object's weight, especially in non-gravitational environments.
• It depends on other forces acting on the object, such as acceleration from a rocket.

The normal force here matches the force due to the rocket's acceleration, calculated to be 735 N. This ensures the crate remains stationary relative to the rocket, even as they both accelerate upward together.

Acceleration

Acceleration is the rate of change of velocity of an object with time. It is a vector quantity, which means it has both magnitude and direction. In this exercise, the rocket accelerates upwards with 9.80 m/s², the same magnitude as Earth's gravitational pull.
Here, the acceleration is not due to gravity, which is absent in the scenario, but due to the force produced by the rocket's engines.

• When objects are in accelerating systems, they can experience forces similar to gravity.
• In the rocket, the upward acceleration mimics the effect of gravity on the crate, causing a normal force equal to weight on Earth.

This understanding allows us to calculate forces acting on objects in such scenarios, helping predict their movements.

Newton's Second Law

Newton's Second Law of Motion comes into play heavily in this situation. It is articulated as \(F = ma\) and describes the relationship between force, mass, and acceleration. In simple terms, it means the force acting on an object is equal to the mass of that object multiplied by its acceleration.
For the crate inside the speeding rocket, we use this key principle:

• Mass of the crate is 75 kg.
• Acceleration due to engines is 9.80 m/s², identical to the gravitational pull on Earth.
• Calculating the force gives \(F = 75 \times 9.80 = 735 \text{ N}\).

This calculation is crucial as it helps determine other forces, such as the normal force, by revealing the impact of acceleration on the crate. This illustrates Newton's law on how forces result in motion changes.