Questions 1, 2, and 3 are short free-response questions that...

Questions 1, 2, and 3 are short free-response questions that require about 13 minutes to answer and are worth 8 points. Questions 4 and 5 are long free-response questions that require about 25 minutes each to answer and are worth 13 points each. Show your work for each part in the space provided after that part. (DIAGRAM IS NOT AVAILABLE TO COPY) A massless tray is placed on an inclined plane with an angle of incline of $\theta$ There is a coefficient of static friction $\mu_{\mathrm{s}}$ between the inclined plane and the massless tray. The tray is attached to a box of mass $M$ by the pulley system shown below. (DIAGRAM IS NOT AVAILABLE TO COPY) (a) If mass can loaded onto the massless tray, how much mass $m$ has to be loaded to stop the tray from being pulled up the inclined plane by $M ?$ (b) How much mass $m$ has to be loaded until the tray starts sliding down the inclined plane? (c) If $\mu_{\mathrm{k}}=0.3$ and $\theta=45^{\circ},$ what is the acceleration of the massless tray if $m=4 M ?$

Questions 1, 2, and 3 are short free-response questions that require about 13 minutes to answer and are worth 8 points. Questions 4 and 5 are long free-response questions that require about 25 minutes each to answer and are worth 13 points each. Show your work for each part in the space provided after that part.
(DIAGRAM IS NOT AVAILABLE TO COPY)
A massless tray is placed on an inclined plane with an angle of incline of $\theta$ There is a coefficient of static friction $\mu_{\mathrm{s}}$ between the inclined plane and the massless tray. The tray is attached to a box of mass $M$ by the pulley system shown below.
(DIAGRAM IS NOT AVAILABLE TO COPY)
(a) If mass can loaded onto the massless tray, how much mass $m$ has to be loaded to stop the tray from being pulled up the inclined plane by $M ?$
(b) How much mass $m$ has to be loaded until the tray starts sliding down the inclined plane?
(c) If $\mu_{\mathrm{k}}=0.3$ and $\theta=45^{\circ},$ what is the acceleration of the massless tray if $m=4 M ?$

Key Concepts

Pulley Systems

Pulley systems are used to connect the motions of different masses and redirect forces. In mechanics, pulleys simplify the study of dynamics by ensuring that forces such as tension are transmitted between objects, which is vital for analyzing how different masses interact and accelerate under gravity and friction.

Free-Body Diagrams

Free-body diagrams are a key analytical tool in mechanics used to visually represent all the forces acting on a body. By isolating an object and mapping out forces like gravity, tension, friction, and the normal force, these diagrams provide clarity and assist in setting up the correct equations for analysis.

Friction

Friction is the resistive force that opposes the motion between two surfaces. The problem involves both static friction, which prevents an object from starting to move until a threshold force is exceeded, and kinetic friction, which acts when an object is already in motion. Understanding these concepts is crucial for determining the conditions under which objects begin to move or maintain equilibrium.

Force Decomposition on Inclined Planes

When dealing with inclined planes, it is essential to decompose forces into components that are parallel and perpendicular to the incline. This decomposition helps in analyzing how gravitational force influences motion along the plane and how the normal force, which affects friction, is determined.

Newton's Laws of Motion

Newton's Laws provide the foundation for analyzing the forces acting on objects in motion or at rest. In these problems, Newton’s second law (F = ma) is used to set up force balance equations that determine whether an object remains in equilibrium or accelerates, considering all applied forces including tension, gravity, and friction.

Transcript

00:01 We're going to be working with an example of two masses joined together on an incline plane via a pulley, and we'll make sure that that's a massless, frictionless pulley, and we'll put a little m on the incline and a large m hanging down from the pulley.

00:27 And we're going to ask, especially if there's two types of friction possible, both static and kinetic that take place on the inclined surface.

00:40 So we'll assume that that's rough enough that you have both types of friction present, both static and kinetic.

00:47 And we'll ask what types of motion are possible? well, first of all, you could have equilibrium.

00:59 That's an interesting situation because even in equilibrium where the acceleration is zero, so you have to worry about whether there's a tendency for the mass on the incline to be either sliding up or sliding down.

01:16 And remember that the static friction can act in either direction.

01:21 So we'll do a free body diagram in this case of the system, both with the case that the little m is trying to slide down.

01:35 And it's also trying to slide up.

01:45 So we'll kind of split up our screen into two different places.

01:50 So with it trying to slide down, let's do a free body diagram.

01:55 I'm going to make that a nice clear dot for little m.

02:02 Kind of just show some angles.

02:05 So we have the weight, mg, going down, making an angle theta with respect to the vertical, or normal to the plane, i should call it.

02:18 There's the normal force, f sub n.

02:26 And there is friction, the static friction in this case, will actually act up the plane.

02:35 And we also have the tension in the cable connecting the two.

02:43 And what's easy about the equilibrium situation is that our big m is being held in equilibrium.

02:52 So it's got m .g.

02:55 Pointing down.

02:56 And there's this interactive force, f tension, which has to equal big mg, if we're in equilibrium.

03:09 Okay.

03:10 But if we're sliding up the plane, if there's a tendency to slide up the plane, then the only thing that's going to change in our force diagram is that the static friction will actually act opposite.

03:26 In the opposite way.

03:28 So we still have mg downwards, still making a angle theta with respect to the normal.

03:38 We still have the normal force and we still have the tension.

03:48 But what will happen is that what will happen? static friction will be going the opposite direction.

04:05 So really interesting situation.

04:06 And we can write our equilibrium conditions if we assume that we're looking at the situation with the maximum static friction, which is equal to mu s times the normal force.

04:23 The normal force is going to be countering the weight component perpendicular to the incline, which is mg cosine theta.

04:37 So let's write our two equations for equilibrium.

04:44 So this is, again, equilibrium, meaning that you have static friction holding things in place, and the tension in the connecting rope is identically equal to the big m times g.

05:00 And let's see.

05:01 So if we're going down the incline, we'll probably want to call the positive direction down, where in the up situation, we'll call the positive direction.

05:10 Direction up, sure.

05:13 So down the plane, we have m -g -sign theta.

05:18 Up the plane, we have f -s, which we'll assume is the maximum, and we'll have the tension, which is equal to big m times g.

05:35 And all those add up to zero, m .a.

05:39 Equals to zero.

05:41 And we could solve this for the mass needed to make things start to slide down the incline.

05:51 If we do that, let's see, we have m .g...

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