Acceleration of a Car on a Frictionless Ramp

[Jacob87411]

I have a question dealing with acceleration. A car is on a frictionless ramp. The car is tied to a string that goes over a pulley and over the pulley there is a hanger. As the mass of the hanger increases should the acceleration of the car increase, decrease or stay the same. I thought it would remain the same since the hangar will accelerate at the same rate regardless of mass but results show otherwise. Thanks

 

[topsquark]

I have a question dealing with acceleration. A car is on a frictionless ramp. The car is tied to a string that goes over a pulley and over the pulley there is a hanger. As the mass of the hanger increases should the acceleration of the car increase, decrease or stay the same. I thought it would remain the same since the hangar will accelerate at the same rate regardless of mass but results show otherwise. Thanks

Think about Newton's 2nd. The greater the mass on the hanger, the greater the force the Earth exerts on it. So a greater mass on the hanger should produce a greater force on the car (via the tension force). Thus the car accelerates faster.

-Dan

 

[Doc Al]

Just to add to Dan's comments...

I thought it would remain the same since the hangar will accelerate at the same rate regardless of mass but results show otherwise.

I presume you are thinking that all objects have the same acceleration due to gravity regardless of mass. That's only true for objects in free fall, where the only force acting on them is gravity. But the hanger is not in free fall; gravity is not the only force acting on it--the string is also pulling it upwards.

 

[Jacob87411]

Ok I have a question then when solving for the forces. First the forces of the hanger that's falling

The sum of the forces = T-mg=ma assuming down is negative, up is positive.

Sum of forces on the car being dragged in the Y direction is 0. Assuming this is frictionless is the sum of the forces just the force of tension pulling the car? Thanks

 

[lightgrav]

yes.
If friction is zero, the only (horizontal) Force applied to the car is Tension.

Be sure to use the correct prepositions :
We add the Forces APPLIED TO the hanger by other things ... these
are called Forces ON the hanger BY the other things (not "of" the hanger).

The sum of the Force y-components on the car is zero ...
this is not quite the same as "sum of forces on the car ... is 0" .

By the way, why don't you call the coordinate along the string as "x"?

 

[Jacob87411]

Well I am trying to derive the equation for acceleration to help explain why it increases and I am having problems doing that.

For the hanger:
Sum of Forces X = 0
Sum of Forces Y = T-mg=ma

Car:
Sum of forces X= T=ma
Sum of forces Y= 0

I know that much but I am having issues deriving for the acceleration of the system to explain why mass increase results in acceleration increase

 

[Doc Al]

Careful with those equations. I see several problems:

(1) You failed to distinguish between the two masses. Call them by different symbols: m_h versus m_c.

(2) You failed to use a consistent sign convention and properly reflect the acceleration constraint. The constraint is that if the car has acceleration +a (up the ramp) then the hangar will have acceleration -a (down).

(3) You neglected the weight of the car.​

Try rewriting those equations with that in mind.