A 30.0 -g metal ball having net charge Q=5.00 μC is thrown out of a window horizontally north at

A 30.0 -g metal ball having net charge Q=5.00 μC is thrown...

A 30.0 -g metal ball having net charge $Q=5.00 \mu \mathrm{C}$ is thrown out of a window horizontally north at a speed $v=20.0 \mathrm{m} / \mathrm{s}$ . The window is at a height $h=$ 20.0 $\mathrm{m}$ above the ground. A uniform, horizontal magnetic field of magnitude $B=0.0100 \mathrm{T}$ is perpendicular to the plane of the ball's trajectory and directed toward the west. (a) Assuming the ball follows the same trajectory as it would in the absence of the magnetic field, find the magnetic force acting on the ball just before it hits the ground. (b) Based on the result of part (a), is it justified for three-significant-digit precision to assume the trajectory is unaffected by the magnetic field? Explain.

Key Concepts

Projectile Motion

Projectile motion is the study of the motion of objects that are launched into the air and influenced only by gravity (in the absence of other significant forces). It involves analyzing the independent horizontal and vertical components of the motion, where the horizontal motion is at constant velocity while the vertical motion is uniformly accelerated motion due to gravity.

Lorentz Force (Magnetic Force on a Moving Charge)

The Lorentz force describes how charged particles experience a force when moving through a magnetic field. This force is given by the equation F = q(v × B), meaning that the magnitude of the force depends on the charge, the speed of the particle, and the magnetic field strength, and it acts perpendicular to both the velocity of the particle and the magnetic field.

Cross Product and Directionality

The cross product in the Lorentz force equation indicates that the direction of the magnetic force is perpendicular to both the velocity of the charged object and the magnetic field. This concept is crucial for predicting how the path of a moving charge will be altered in a magnetic environment, and it is determined using the right-hand rule.

Order-of-Magnitude Analysis and Significant Figures

Order-of-magnitude analysis involves estimating the relative sizes of different forces or effects to decide if some contributions can be neglected. When high precision (such as three significant digits) is required, it is important to assess whether minor forces, like a weak magnetic force compared to gravitational force, are significant enough to alter the expected trajectory.

Transcript

00:01 A ball with the mass of 30 grams and a charge of 5 microcouons is thrown out of a window with the height of 20 meters and also with the velocity of 20 meters per second.

00:13 And there's also a magnetic field of 0 .01 tesla's that is perpendicular to the trajectory of the ball.

00:25 So we're going to start solving part a where it asks us to find what the force on the ball is due to the magnetic field.

00:33 Right before it hits the ground.

00:39 So we're going to start by saying that the velocity in the y direction is equal to the square root of 2 times g times h.

00:54 And we can calculate this velocity by plugging in g, which is 9 .8 meters per second squared, and then plugging in h, which is given as 20 meters.

01:13 And so this will give us a velocity that's equal to 19 .8 meters per second.

01:23 And now we can find the magnitude of this velocity, which i'll call v.

01:32 I'll call it v of r.

01:35 And this is going to be equal to the square root of the velocity in the x direction squared plus velocity in the y direction squared.

01:50 So this v of r, you can find by plugging in v of x direction, which is given to us by the problem as 20 meters per second.

02:07 And then we can also plug in v of y, which we just calculated.

02:16 And so this will give us a magnitude that is equal to 28 .1 meters per second.

02:28 And now the force, due to the magnetic field, we can find sq times v...

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