II It's moving day, and you need to push a 100 kg box up a...

II It's moving day, and you need to push a $100 \mathrm{~kg}$ box up a $20^{\circ}$ ramp into the truck. The coefficients of friction for the box on the ramp are $\mu_{\mathrm{s}}=0.90$ and $\mu_{\mathrm{k}}=0.60 .$ Your largest pushing force is $1000 \mathrm{~N}$. Can you get the box into the truck without assistance if you get a running start at the ramp? If you stop on the ramp, will you be able to get the box moving again?

Key Concepts

Kinetic Friction

Kinetic friction acts between moving surfaces and is typically lower than the maximum static friction. Once motion has begun, kinetic friction determines how much force is required to maintain or change the speed of an object. This concept is important for analyzing the dynamics of motion and the energy losses during movement.

Static Friction

Static friction is the force that resists the initiation of motion between two surfaces in contact. It has a maximum threshold that must be overcome for movement to occur. Understanding static friction is critical when considering whether a force is sufficient to start moving an object at rest, as it directly determines the onset of motion.

Newton's Second Law

Newton's Second Law establishes the relationship between net force, mass, and acceleration (F = ma). This principle is fundamental in analyzing whether a given applied force can overcome resistive forces like friction and gravity, thereby causing an object to accelerate. It provides the framework for evaluating the motion of objects under various force conditions.

Inclined Plane Dynamics

This concept focuses on understanding how objects behave on a sloped surface. It involves analyzing the components of gravitational force that act parallel and perpendicular to the surface, determining how the incline affects normal forces and overall acceleration. Understanding inclined plane dynamics is crucial for solving problems where external forces, such as pushing or pulling, are applied on slopes.

Force Decomposition

Force decomposition is the process of breaking a force into its components along different axes, typically parallel and perpendicular to an inclined surface. This allows for a better analysis of the various forces acting on the object, including gravitational, normal, and applied forces. By resolving forces, one can precisely assess the net force that causes acceleration or deceleration.

Transcript

00:01 In this question, we're told that a 100 -kilogram box is pushed up a 20 -degree ramp into a truck.

00:08 Co -efficient of friction on the ramp are 0 .9 for static and 0 .6 for kinetic.

00:15 Your largest pushing force is 1 ,000 newtons.

00:17 And the question is, can you get the box into the truck without assistance if you get a running start at the ramp? so in order to do this question, we need to do a free body diagram.

00:31 And look at the forces to see if our pushing force can overcome the other forces that are pulling the box down the ramp.

00:45 So let's start with a free body diagram.

00:48 So this will be our box.

00:50 And of course there's a normal force that is directed perpendicular to the ramp.

00:55 There's going to be the applied force.

00:58 That's a thousand newtons pushing up the ramp.

01:01 And then we will also have a gravitational force pulling straight down.

01:06 And so a component of that is along the ramp as well.

01:10 So we're going to need to be able to overcome that as well as the friction force.

01:16 Now the direction of the friction force in this case is going to be down the ramp because it's going to be opposite of the intended motion.

01:26 So in order to find out if we can move this box, we need to figure out if the applied force can overcome both the friction force and the gravitational force.

01:42 So what i would suggest doing, and this is a static friction force because the box isn't moving yet, what i would suggest doing is calculating the gravitational force as well as the friction force.

01:55 Now, the friction force, the static friction force can be anywhere between zero and some maximum value.

02:09 So the person is going to have to overcome the maximum amount of static friction, because static friction will increase, increase, increase, increase, until you can overcome it, until you can overcome that maximum amount that it takes on.

02:24 So the maximum amount of static friction is going to be mu s times the normal force.

02:31 The normal force in this case, we can look at the, do a quick analysis of the y direction here.

02:37 The normal force is going to be completely balanced by the gravitational force in the y direction.

02:45 So those two things are equal.

02:47 You can replace fn with fgy.

02:51 And just for reference, this is the coordinate system that i'm using in the problem.

03:00 So the component of the gravitational force that is in the y direction is equal to mg, cos 20 degrees...

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