How Does a Block's Motion Change on an Inclined Ramp with Friction?

[justbreathe87]

Hi! This is my first post, so if I do anything incorrectly I apologize

Homework Statement

A block of wood slides first down and then up a ramp inclined at 25°. Assume the block/ramp kinetic friction coefficient = 0.2. The block starts from rest at position located 3.2 m up from the lowest point on the ramp. After sliding down, the block bounces back at the lowest point. The speed just after the bounce is only 87% of what it was just before the bounce.

a) Draw the free-body diagram for the block during the downward motion.
b) Using the above diagram, find the block's acceleration on the way down.
c) How much time does the block travel until it hits the lowest point?
d) Draw the free-body diagram for the block during the upward motion.
e) Using this diagra, find the block's acceleration on the way up.
f) Find the time the block spends moving up after the bounce until it reaches the highest point.
g) How far is that point from the bounce-point?

Homework Equations

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The Attempt at a Solution

a) I have the FBD drawn correctly, no help needed here.

b) a = -g sin(25°) - 0.2cos(25°)
a = -2.37 m/s^2 (I am pretty sure this is correct as well)

c) time = square root of 2d/a = square root of (2(3.2)/2.37)
t = 1.64 seconds (not feeling good about this part)

d) No help needed on the FBD.

e) -g sin(25°) + 0.2cos(25°)
a = -5.92 m/s^2

f) velocity = t * |a| = 1.64 * 2.37 = 3.89 m/s
3.89 * 82% = 3.19 m/s
t = v/|a| = 3.19/5.92 = 0.54 seconds

g) distance = v*t
d = (3.19)(0.54) = 1.72 m

I don't feel like I did c, f, or g correctly. Sorry this was so long and probably hard to read. I am just getting used to this! Any help would be appreciated, even if it is only for one of the letters. Thank you!

 

[kuruman]

Hi justbreathe87. Welcome to PF.

a) I have the FBD drawn correctly, no help needed here.

b) a = -g sin(25°) - 0.2cos(25°)
a = -2.37 m/s^2 (I am pretty sure this is correct as well)

It is not correct for the following reasons:

1. It is dimensionally incorrect. The first term has (correctly) dimensions of acceleration, but the second term does not.

2. When the block is sliding down the incline, the component of the acceleration g sin(25^o) is in the direction of motion (down the incline) but the second term which is due to friction is in the opposite direction (up the incline). The two terms then must have opposite algebraic signs. If one is positive, the other one must be negative. You have written them both with the same sign.

Once you have the correct value for the acceleration you can proceed to the other parts.

Part (c) the method is correct.
Part (e), again you need to watch your dimensions and the relative signs. Here the two terms of the acceleration must have the same relative sign, because they both point down the incline when the block is moving up the incline.
Part (f), the method is correct
Part (g), you need to use the kinematic equation to find the distance. You know the initial velocity, you know the acceleration and you know the time. The equation
distance = v*t is valid only when the acceleration is zero.

 

[kerimek]

Hi there! It looks like you have a good understanding of the problem and have made some correct calculations. I will provide some feedback and additional information to help you improve your solution.

a) Your free-body diagram should include all the forces acting on the block, which are gravity, normal force, and friction force. Make sure to include the direction and magnitude of each force.

b) Your calculation for acceleration is correct. Just remember to include the units, which in this case would be m/s^2.

c) Your calculation for time is incorrect. The formula you used is for calculating the time it takes for an object to fall a certain distance, but in this case, the object is sliding down a ramp. You can use the formula t = sqrt(2d/a) to find the time it takes for the block to slide down the ramp from its initial position to the lowest point. However, you also need to consider the time it takes for the block to bounce back up to its initial position. To find this time, you can use the formula v = u + at, where u is the initial velocity (zero in this case) and v is the final velocity (which is 87% of the initial velocity). You can then use the formula t = (v-u)/a to find the time. Once you have both times, you can add them together to get the total time.

d) Your free-body diagram is correct.

e) Your calculation for acceleration is correct.

f) Your calculation for time is incorrect. You can use the formula t = v/a to find the time it takes for the block to move from the lowest point to the highest point. However, you also need to consider the time it takes for the block to bounce back down to the lowest point. To find this time, you can use the formula v = u + at, where u is the initial velocity (87% of the velocity at the highest point) and v is the final velocity (zero in this case). You can then use the formula t = (v-u)/a to find the time. Once you have both times, you can add them together to get the total time.

g) Your calculation for distance is incorrect. You can use the formula s = ut + 1/2at^2 to find the distance traveled from the bounce point to the highest point. However, you also need to consider the distance traveled from the highest point to the bounce point