A small block of mass m and charge Q is placed on an...

Key Concepts

Static Equilibrium

Static equilibrium refers to the state where an object remains at rest because the sum of all forces acting on it is zero. In problems involving equilibrium, especially on inclined planes, it is essential to consider all the forces, including gravitational, normal, frictional (if applicable), and any additional forces like electric or magnetic forces, to ensure that the net force in every direction equals zero.

Force Decomposition on an Inclined Plane

When analyzing objects on an inclined plane, gravitational force is commonly broken down into components that are parallel and perpendicular to the plane. The parallel component (mg sin?) influences the motion along the incline, while the perpendicular component (mg cos?) is counteracted by the normal force. This decomposition is a fundamental step in solving equilibrium and dynamics problems on inclines.

Electric Force on a Charged Object

The electric force acting on a charged object is given by the product of the charge and the electric field (F = QE). This concept is crucial when considering the influence of an electric field, especially when it acts along the same direction as other forces such as the gravitational force on an inclined plane. The sign of the charge determines the direction of the force relative to the electric field.

Net Force and Force Balance

For an object to remain stationary, the net force acting on it must be zero. This involves setting up an equation where all the forces along a specific direction (usually the direction of potential motion) sum to zero. Achieving force balance might involve counteracting gravitational components with electric or other forces, which is a central idea in solving static equilibrium problems.

Transcript

00:01 So according to problem 31, there's a block of mass m and charge q in a certain electric field not specified yet.

00:12 Okay.

00:14 And this block of mass is positioned on a frictionless inclined plane of angle theta.

00:21 Okay.

00:23 And we've been asked to find out the magnitude of the electric field.

00:31 Such that the block is addressed.

00:34 Okay, this is the first task, and the second task is very specific numbers for mass, for charge, and for the angle there.

00:47 And we need to determine the magnitude of the electric field in numbers, okay, in the direction of the electric field specifically.

00:55 Okay, let's go on with the solution of this problem.

00:58 The very first thing we need to do is um yeah it's draw the um it's draw the forces excited on this body m and of course the very first yeah immediately the first force that comes to mind is the weight okay that points down that's also and that's also the reaction from the ground that's called it t t vector and now there should be another another another another force that will be the electric force the electric force let's call it f electric okay and what's the relation between the electric force and the electric field this is the one is the formula the electric field vector that field equals the charge times the vector oh excuse me the electric force equals the charge times the vector the electric field okay now we've been very specifically given that the body is at best therefore the sum of forces equals zero meaning therefore if we we do the vector sum take the vector sum of the weight plus the reaction to the ground and the undetermined yet electric force that will be zero okay and of course we understand that they don't know the point they point towards different directions and we understand that this t might depend actually on on the electric force but there's a way we can make it such that t does not depend on the electric force if we take the electric force to be to be perpendicular to t, the reaction to the ground, or parallel to the inclined plane.

03:07 This way, and this is quite crucial, this way, it might be easy to see that this specific component of the weight, okay, if, if there's angle theta, therefore it might not it might be easy to see that it is also angle theta and okay therefore the the reaction to the ground as a as a magnitude is countered by m g the magnitude the magnitude of the of the weight times cosine cosine okay therefore m times g sine theta meaning the other component of the weight if we decompose the weight that specific component of the weight is the one that counters the effect of the electric force in this way i have this is the symbol okay, for the magnitude of the electric force.

04:41 Therefore, since we know basically everything, we know mass, we know g the acceleration of gravity, we know angle theta, we know charge, and of course the magnet of the electric force will be equal to the absolute value of the charge times the magnitude of the electric force, excuse me, electric field...

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