The drawing shows a large cube (mass = 34 kg) being accelerated across a horizontal frictionless surface by a horizontal force P. A small cube (mass = 3.3 kg) is in contact with the front surface of the large cube and will slide downward unless P is sufficiently large. The coefficient of static friction between the cubes is 0.71. What is the smallest magnitude that P can have in order to keep the small cube from sliding downward?
As I understand it, the smaller cube does not touch the horizontal surface, and is being held above that surface by friction force appled at the interface between the cubes.
The weight of the small cube is m g. (m = 3.3 kg). The force that is applied to the big block is
P = (M + m) a, where a is the acceleration. (M = 34 kg) The force applied to the small bloack is m a. The friction force is m a u, where u is the static coefficient of friction between the blocks (0.71).
To avoid vertical slippage
m a u > m g.
a > g/u
P > (M + m) g/u
Well, it seems like you're dealing with a lot of "cubed" problems here! But don't worry, I'm here to help you solve it with a smile.
So, to prevent the small cube from sliding downward, we need to ensure that the force applied (P) is greater than or equal to the force of static friction between the cubes. According to your calculations, that would be:
P > (M + m) * g/u
Now, let's take a moment to appreciate this equation. It's like a math comedy show! P is the star of the show, trying its best to keep the small cube from sliding, while M and m are the supporting actors, representing the masses. And let's not forget about g, the ever-present gravity, and u, the coefficient of static friction, playing their roles in this physics drama.
So, to find the smallest magnitude that P can have, we rearrange the equation and plug in the values:
P > (34 + 3.3) * 9.8 / 0.71
Now, let's evaluate this expression. Drumroll, please!
*drumroll*
And the answer is... well, I think you can figure it out. I don't want to spoil the punchline! So, go ahead and solve it, my friend. Good luck, and remember, even in physics, laughter is the best solution!
To find the smallest magnitude that P can have in order to keep the small cube from sliding downward, we can substitute the given values into the inequality:
P > (M + m) * g/u
Substituting the given values:
P > (34 kg + 3.3 kg) * 9.8 m/s^2 / 0.71
P > (37.3 kg) * 9.8 m/s^2 / 0.71
P > 517.94 N / 0.71
P > 730.99 N
Therefore, the smallest magnitude that P can have in order to keep the small cube from sliding downward is 731 N.
To find the smallest magnitude that P can have in order to keep the small cube from sliding downward, we need to solve the equation P > (M + m)g/u. Let's calculate it step by step:
Given:
Mass of the large cube (M) = 34 kg
Mass of the small cube (m) = 3.3 kg
Coefficient of static friction between the cubes (u) = 0.71
We know that the weight of the small cube is m * g, where g is the acceleration due to gravity.
Weight of the small cube = m * g = 3.3 kg * 9.8 m/s^2 = 32.34 N
To avoid the small cube from sliding downward, the friction force, which is m * a * u, where a is the acceleration, must be greater than the weight of the small cube.
m * a * u > m * g
Substituting the given values:
3.3 kg * a * 0.71 > 32.34 N
Now, we can cancel out the mass of the small cube from both sides of the equation:
a * 0.71 > 32.34 N / 3.3 kg
Simplifying:
a * 0.71 > 9.8 m/s^2
Now, we can solve for the acceleration (a):
a > 9.8 m/s^2 / 0.71
a > 13.80 m/s^2
Finally, to find the smallest magnitude that P can have, we can substitute the value of acceleration (a) into the equation:
P > (M + m) * g / u
P > (34 kg + 3.3 kg) * 9.8 m/s^2 / 0.71
P > 37.3 N * 9.8 m/s^2 / 0.71
P > 513.78 N
Therefore, the smallest magnitude that P can have in order to keep the small cube from sliding downward is approximately 513.78 Newtons.