Conservation of Energy-Two Blocks Sliding Down an Incline

[_N3WTON_]

Homework Statement

Block 1 and Block 2 have the same mass 'm', and are released from the top of two inclined planes of the same height making 30 degree and 60 degree angles with the horizontal direction, respectively. If the coefficient of friction is the same in both cases, which of the blocks is going faster when it reaches the bottom of its respective incline?

Homework Equations

mgh = 1/2mv^2

The Attempt at a Solution

Using the above equation, I found that both blocks would have the same speed once they reached the bottom of the incline. However, block 2 would reach the bottom first due to it's PE being converted into KE faster than block 1. However, my instructor informed me that my answer "Both blocks have the same speed at the bottom" is incorrect and in fact Block 2 is faster at the bottom. Can anyone explain why this is the case? Thanks.

 

[PhanthomJay]

Your energy equation neglected the work done by friction. They would have the same speed if there was no friction. That is not the case here.

 

[_N3WTON_]

thank you

 

[Rellek]

In case you need a refresher for work and energy equations:

$$T_{1} + U_{1} + \int{Fds} = T_{2} + U_{2}$$

Where $T$ and $U$ are your kinetic energy and gravitational potential energy, respectively.

If the force applied to your object is constant, as is the case with friction, work can simply be defined as force multiplied by the distance over which it is implied.

Using trigonometry, how could you compare the sliding distances of the two blocks, keeping in kind that the heights that the blocks start out at are the same?

I suggest drawing out this scenario with the equivalent forces to help you out.

 

[HalfLostAndSearching]

I would first like to clarify that your initial attempt at a solution using the conservation of energy equation is correct. Both blocks would have the same speed at the bottom of their respective inclines. However, it is important to consider the factors that could affect the speed of the blocks, such as air resistance and the angle of the incline.

In this scenario, the coefficient of friction is the same for both blocks, meaning that the force of friction acting on each block is equal. However, the angle of the incline does play a role in determining the speed of the blocks. Since block 2 is on a steeper incline, it will experience a greater component of gravity pulling it down the incline, resulting in a greater acceleration and therefore a higher speed at the bottom.

Additionally, air resistance could also have an impact on the speed of the blocks. Since block 2 is on a steeper incline, it will have a shorter distance to travel and therefore less time for air resistance to act upon it, resulting in a slightly higher speed at the bottom compared to block 1.

In conclusion, while both blocks may have the same speed according to the conservation of energy equation, factors such as the angle of the incline and air resistance can affect the actual speed of the blocks. It is important to consider all factors in a real-world scenario and not rely solely on theoretical equations.