Calculate the magnitude of the force required to give a 0.145-kg baseball an acceleration a = 1.

step 1 This is chapter 37, problem number 26. We're given the mass of baseball, and we are given the ac

Calculate the magnitude of the force required to give a...

Key Concepts

Newton's Second Law

This fundamental principle of classical mechanics states that the force acting on an object is equal to the mass of the object multiplied by its acceleration (F = ma). It is normally applied when dealing with speeds much less than the speed of light and forms the basis for many introductory problems in dynamics.

Special Relativity

When objects move at speeds comparable to the speed of light, classical mechanics gives way to special relativity. In this framework, the formulas for momentum, energy, and force are modified due to effects such as time dilation and length contraction, meaning that simple F = ma no longer holds in its usual form.

Lorentz Factor

The Lorentz factor, denoted by ? (gamma), quantifies how much time, length, and relativistic mass change for an object moving at a significant fraction of the speed of light. It is critical in calculating corrections to Newton's equations at high speeds and plays a central role in determining how forces and accelerations transform in special relativity.

Relativistic Dynamics: Directional Dependence

In special relativity, the effect of a force on an object depends on the direction of the applied force relative to the object's velocity. For forces acting parallel to the velocity, the effective inertial mass increases with a factor of ?³, while for forces applied perpendicular to the velocity, it increases with a factor of ?. This directional dependence is essential for accurately calculating the forces required to produce a given acceleration at relativistic speeds.

Transcript

00:01 This is chapter 37, problem number 26.

00:03 We're given the mass of baseball, and we are given the acceleration, although baseball we want to be one meters per second square.

00:13 And for a series of initial speeds that are given to us in part a, b and c, part being 10 meters per second, b being 0 .9c, and 0 .99c.

00:24 We were asked to calculate the force that it takes to accelerate this baseball to the given value here.

00:32 Now, since part a is not dealing with the relativistic velocities, f is really straightforward, right, it's m times a.

00:39 So m is 1 .4 .5 kilograms, and the acceleration is one meter per second square.

00:49 So as an answer to part a, we have 0 .145 mutants.

00:55 Now in part b, as you can see, now the speed is much higher.

01:01 So let's answer part b here.

01:04 When that's the case, on your textbook, you can go and find equations 37 .32 and 37 .33.

01:17 You're going to figure out that the relativistic force is going to be depending on whether or not the acceleration is, in the direction of the velocity vector or it is perpendicular with respect to the velocity vector.

01:33 We have different equations for each.

01:37 So if f and v vector are along, along the same line, then we have f equals cube of gamma factor times a.

01:54 If f and v are perpendicular then we actually have gamma vector m times a so in part b since it's given to us that they're in the same line then this is the equation that we're going to use for both part b and c okay so going back to part b now i'm asser it's part b would be f equals gamma cubed m times a now, the gamma factor, as you know, 1 over square root of 1 minus v, in this case, is 0 .9c squared, over c squared.

02:38 But we're going to take the cube of this.

02:41 1 is 1.

02:43 And we're going to take the cube of denominator then m is going to be 0 .145 kilograms, and the acceleration is again 1 meters per second square.

02:53 So if you do everything correctly, then the force.

02:56 It's going to be found as 1 .75 newtons.

03:01 Now, we're going to do the exact same thing for part c, still where the acceleration and the velocity are along the same line, which means the force is going to be along the same line as the velocity vector.

03:14 So we're going to use the same equation...

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