Hanging mass pulling another mass

[lightng]

This is question 89 of the first Physics Question Pack

Basically, there are 2 masses. 1 is on a wooden board and the other one is hanging off of such board. Both masses are linked by a string that is passed over a pulley.

When sufficient mass is hanging off the board, it causes the mass on the board to slide.

The question was:

"After a block began to slide, how did its speed vary with time? (Note: Assume that the tension and kinetic friction forces on the block were constant in magnitude.)"

I reasoned that it would increase exponentially with time because I thought I could equate the gravitational potential of the "hanging mass" to transfer to kinetic energy of the mass that is being pulled.

However, the reasoning is that the speed increases linearly with time".

AAMC reason:

"The coefficient of kinetic friction is always lower than that of static friction. Therefore there is a net accelerating force on the block once it starts to slide. A constant force on a mass produces a constant acceleration (Newton's second law). Thus, the velocity of the block increases linearly with time.



Can someone explain to me why my logic was flawed or why I came to the incorrect conclusion?

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[theonlytycrane]

The masses are connected by a string so they will move together at the same speed. If the mass on the board begins to slide, there must be a net force on the hanging mass (down), accelerating the blocks. Velocity is related to acceleration by: v = v(initial) + a * t, so the velocity will increase linearly with respect to time (t).

Assuming there's no friction, the potential energy of the system decreases and the kinetic energy of the system increases as you stated. But the energy exchange is linear with respect to time based on the velocity / acceleration relationship above, and not exponential.

 

[BerkReviewTeach]

You can get this from the kinematics equation vt = at + vo. It's a linear relationship between vt and t if a is constant. Every second, the speed goes up by a set amount. Let's say a were 3.0 m/s2 for this example. At t = 0, v = 0. At t = 1, v = 3. At t = 2, v = 6. If you graph this, you'll see it's a line.

 

[LIC2015]

I reasoned that it would increase exponentially with time because I thought I could equate the gravitational potential of the "hanging mass" to transfer to kinetic energy of the mass that is being pulled.

The only net force here is the one exerted by the suspended mass, M1. If you think about the value and units of gravity (acceleration), they are 10 m/s^2. If you think about this intuitively, these units state that for every second that goes by, an object that has only gravity acting on it will increase in speed by 10 m/s for every second that goes by, neglecting air resistance. If you track the speed of a falling object like M1 (which has the same speed as M2), you can say that after 1 second of falling, the mass is moving at 10 m/s. After another second goes by, it'll be falling at 20 m/s. After another second, it'll be falling at 30 m/s, and so on, until it reaches terminal velocity. Velocity does not increase or decrease exponentially when only gravity is acting on it. It always changes linearly. Make sense?

 

[BerkReviewTeach]

The only net force here is the one exerted by the suspended mass, M1. If you think about the value and units of gravity (acceleration), they are 10 m/s^2. If you think about this intuitively, these units state that for every second that goes by, an object that has only gravity acting on it will increase in speed by 10 m/s for every second that goes by, neglecting air resistance. If you track the speed of a falling object like M1 (which has the same speed as M2), you can say that after 1 second of falling, the mass is moving at 10 m/s. After another second goes by, it'll be falling at 20 m/s. After another second, it'll be falling at 30 m/s, and so on, until it reaches terminal velocity. Velocity does not increase or decrease exponentially when only gravity is acting on it. It always changes linearly. Make sense?

As a point of clarity, there are two forces acting on M1, mg down and T up. As such, it is not in free fall, so the actual acceleration, a, is less than g. You have a good explanation here, but I wanted to add that a will be less than 10, in case someone mistakenly thinks you are saying that it's 10 in this case (as opposed to your hypothetical scenario).