A mug rests On an inclined surface, as shown in FlGURE 6-35 . (a) What is the magnitude of the f

step 1 Okay, let's summarize the problem first. So we have an inclined surface with an angle, an inclin

A mug rests On an inclined surface, as shown in FlGURE 6-35...

Key Concepts

Coefficient of Static Friction

The coefficient of static friction is a dimensionless number that represents the ratio of the maximum static frictional force that can be exerted between two surfaces to the normal force. It quantifies the grip or resistance against sliding, determining the conditions under which an object on an inclined plane will start to move.

Inclined Plane

An inclined plane is a flat surface tilted at an angle, used in physics to analyze how forces act on objects. It is especially useful for breaking down gravitational force into components that are parallel and perpendicular to the surface, facilitating the study of motion or equilibrium on a slope.

Force Decomposition

Force decomposition is the process of breaking a force into several components, typically along mutually perpendicular axes. In the context of an inclined plane, it involves resolving gravity into one component parallel to the slope (which drives the motion) and another component perpendicular to the slope (which influences the normal force).

Normal Force

The normal force is the perpendicular contact force exerted by a surface on an object in contact with it. For an object on an inclined plane, the normal force is generally less than the full weight of the object, and it plays a critical role in determining the magnitude of frictional forces.

Static Friction

Static friction is the force that resists the initiation of sliding motion between two surfaces in contact. It adjusts in magnitude to counteract applied forces up to a maximum value, which is determined by the coefficient of static friction and the normal force, thereby keeping an object at rest on an inclined plane.

Transcript

00:01 Okay, let's summarize the problem first.

00:06 So we have an inclined surface with an angle, an incline angle, and we call it theta, and we have a mug on the surface with a mass m equals 0 .27 kilogram.

00:25 It's always easier to first work with these symbols for the physical quantities instead of actual numbers and only plug in the actual numbers in the end of your calculation.

00:40 So the next thing is to do a free -body diagram analysis on this object, the mark.

00:50 So we know that it must be feeling gravity.

00:56 So the magnitude of gravity would be m -g.

01:00 And it also will feel a normal force perpendicular to the surface on the outward.

01:13 So what else? what other forces could it feel? and from the question of the problem, for example, part a is asking about the magnitude of fictional force.

01:27 We know that it must feel a friction.

01:30 But also from the physical principles, we know that other than the physical principles, we know that other other than these two forces, it must be feeling a force along the direction of the surface.

01:45 Otherwise, this mug wouldn't be in equilibrium.

01:49 It will be accelerating along the surface.

01:52 And we actually know that force must be pointing upward along the surface.

01:58 And really, there's no other options but a friction.

02:03 And also because this mug is at rest on the surface, this friction should be static friction.

02:11 So we have drawn the free body diagram.

02:17 And another thing we can do before going into solving equations is to decompose gravity onto two directions.

02:28 Let's call this x and call this y.

02:32 And from our geometry class, we would know that this angle is also theta.

02:43 And we see that the x component of gravity should be m g sine theta, mg sine theta.

02:57 And the y component of gravity should be m g cosine theta.

03:06 Mg coson.

03:07 So we could replace this gravity with its two components on x and y x -exes.

03:19 Now maybe for clarity, let me erase this gravity force we join in the first place.

03:29 Now, we only have variabilities to components.

03:35 So let's look into the problem.

03:38 Part a is asking about the magnitude of this friction.

03:44 Magnitude of friction.

03:46 How do we find that? now, for this problem, which is an equilibrium problem.

03:53 Our principle is newton's first law, which says the total force acting on an object, the equilibrium must be zero.

04:06 Now, if we break the force into two components, we should say that the sum over x component of all forces should be zero, and the sum of y components.

04:19 Component of all forces should be zero.

04:23 Okay, friction is clearly on the x -axis.

04:27 So let's use the first equation here...

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