What Determines the Block's Motion on an Accelerating Wedge?

[zeena2009]

Homework Statement

A 2kg block rests on a frictionless wedge that haas 60 degree incline and an acceleraion a to the right such that the mass remains stationary relative to the wedge.
a) determine a?
b) what would happened if the wedge were given an acceleration larger than this value?
c) what would happened if the wedge were given an acceleration smaller than this value?



I got a, it's b and c i didnt get
i think that when you give larger accelertaion, the block would slides up
and smaller, the blick slides down the incline
that's my common sense

however, when i tried to do it mathmatically and algebrically, it didnt make sense.

so please help me ??

 

[LowlyPion]

Welcome to PF.

Draw a diagram. Consider the interface between the incline and the block.

The inertia of the block will have a force in reaction to the force of the accelerating incline. Viewed in the frame of reference of the incline then the block is subject to the two accelerations - 1 horizontal and the other vertical. You found the answer when they were in balance, but as you note then when the acceleration is greater than equilibrium condition, the component || along the plane will be greater than needed to resist the || component of the vertical gravity.

If you think about it, this is similar to a car on a banked curve. But rather than the incline accelerating, you have the centripetal acceleration from the circular motion.

 

[nlightner]

I would first commend you for trying to solve the problem both intuitively and mathematically. It shows that you are thinking critically about the problem.

Now, let's address the questions.

a) To determine the acceleration, we can use Newton's Second Law, which states that the sum of all forces acting on an object is equal to its mass times its acceleration (ΣF=ma). In this case, the only force acting on the block is its weight (mg), which is equal to the normal force exerted by the wedge (N) since there is no friction. Therefore, we can write the equation as mg = ma. Since the mass of the block is given as 2kg, we can solve for the acceleration a = g. So, the acceleration of the wedge must be equal to the acceleration due to gravity (9.8 m/s²).

b) If the wedge were given an acceleration larger than this value, the block would start to slide up the incline. This is because the force of gravity is no longer balanced by the normal force, and the net force acting on the block would be in the direction of the wedge's acceleration. The block would accelerate up the incline until it reaches a point where the normal force is equal to the force of gravity, and then it would remain stationary relative to the wedge again.

c) Similarly, if the wedge were given an acceleration smaller than the value of g, the block would start to slide down the incline. This is because the normal force would be greater than the force of gravity, causing a net force in the direction of the incline's downward acceleration. The block would continue to accelerate down the incline until it reaches a point where the normal force is equal to the force of gravity, and then it would remain stationary relative to the wedge again.

In summary, the acceleration of the wedge must be equal to the acceleration due to gravity for the block to remain stationary relative to the wedge. Any deviation from this value would result in the block sliding up or down the incline. I hope this helps to clarify the problem for you. Keep up the good work!