Finding Sliding Distance SS = 5.068 m

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In summary, the block initially has a horizontal velocity of approximately 5.4589 m/s after being projected from the spring and descends a ramp with a height of 2.0 m. It then enters a rough section with a coefficient of kinetic friction of 0.30 and slides a distance of approximately 2.3469 m before coming to a stop at point D. The acceleration of the block on the rough section is -2.94 m/s2.
  • #1
bj5110
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A 1.0 kg block is held in place against a spring with spring constant k = 100 N/m (see the figure, below) by a horizontal external force. The spring is compressed 20 cm. The external force is removed, and the block is projected with some (horizontal) velocity upon separation from the spring. The block descends a ramp with height h = 2.0 m. The track is frictionless between points A and B. The block enters a rough section at B, extending to E. The coefficient of kinetic friction over this section is 0.30. The block slides through this rough section a distance S, and comes to a stop at point D.
What is the sliding distance S?

Lokk6AE.png


Can someone please tell me if my below attempt is the correct answer?

Find velocity after reaches bottom of hill (by conservation of mechanical energy)
( 1/2 * k * x2 ) + ( 1/2 * m * g ) = 1/2 * m * v2
(1/2 * 100 * .20) + (1/2 * 1 * 9.8) = 1/2 * 1 * v2
14.9 = 1/2 * v2
v ≈ 5.4589 m/s

Find the acceleration once it hits the rough spot
μmg = Fnet
0.30 * 1 * 9.8 = 2.94 N

2.94 N = m * a
2.94 N = 1 * a
a = -2.94 m/s2

Kinematics to find distance of S
vf2 = vi2 + 2ax
0 = 5.45892 + ( 2 * -2.94 * x)

x = 5.068 m
 
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  • #2
KE bottom of hill -
check - gravitational potential energy, and shouldn't the 0.20m be squared?

Best practice:
- finish the algebra before subbing in the numbers.
i.e. ##-\mu m g = ma\implies a=-\mu g## ... then substitute.

- use the variable names from the problem in your equations.
i.e. the distance slid is S not x.
 
  • #3
Awesome, thank you for the tips and catching my mistake.
Could you review my corrections and let me know if my answer now seems to be correct?

Find velocity after reaches bottom of hill (by conservation of mechanical energy)
( 1/2 * k * x2 ) + ( 1/2 * m * g ) = 1/2 * m * v2
(1/2 * 100 * .202) + (1/2 * 1 * 9.8) = 1/2 * 1 * v2
6.9 = 1/2 * v2
v ≈ 3.7148 m/s

Find the acceleration once it hits the rough spot
-μmg = ma
a = -μg
a = 0.30 * 9.8 = -2.94 m/s2

Kinematics to find distance of S
vf2 = vi2 + 2ax
0 = 3.71482 + ( 2 * -2.94 * S)

S ≈ 2.3469 m
 
  • #4
... check this:
( 1/2 * k * x2 ) + ( 1/2 * m * g ) = 1/2 * m * v2
"1/2 * m * g" is "half the weight" - a force. All the other terms are energy.

You should get used to scanning your equations to see that the dimensions make sense.
Please note: I won't check your arithmetic.
 
  • #5


Yes, your attempt is correct. The sliding distance S is equal to the distance traveled by the block on the rough section, which is 5.068 m. This can be calculated using the kinematics equation vf2 = vi2 + 2ax, where vf is the final velocity (which is 0 m/s as the block comes to a stop), vi is the initial velocity (which is 5.4589 m/s as calculated using conservation of energy), a is the acceleration (which is -2.94 m/s2 as calculated using the force equation), and x is the distance traveled. Solving for x, we get x = 5.068 m.
 

1. What is Sliding Distance SS?

Sliding Distance SS refers to the distance that an object slides or moves over a surface. It is an important measurement in physics and engineering, as it can help determine the speed and force of an object.

2. How is Sliding Distance SS calculated?

Sliding Distance SS is calculated by multiplying the initial speed of an object (S) by the time it takes for the object to come to a stop (t). The formula for sliding distance is: SS = S * t

3. What units are used to measure Sliding Distance SS?

Sliding Distance SS is typically measured in meters (m) or centimeters (cm). In some cases, it may also be measured in feet (ft) or inches (in).

4. Why is Sliding Distance SS important?

Sliding Distance SS is important because it can help determine the frictional force between two surfaces. It is also a key factor in understanding the motion and energy of an object.

5. How can Sliding Distance SS be manipulated?

Sliding Distance SS can be manipulated by changing the initial speed of the object or by altering the surface it is sliding on. For example, increasing the initial speed will result in a longer sliding distance, while increasing the roughness of the surface will decrease the sliding distance.

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