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A horizontal spring has a coefficient of k = 325 N/m and is compressed by a 4.5 kg block to 0.50 m past its rest position. The block is then released and slides across a flat horizontal surface for a distance of 5.0 m. If the coefficient of friction between the block and the surface is 0.3, how fast (in m/s) would the block be moving at the end of the surface?

Sagot :

MathPhys

Answer:

6.9 m/s

Explanation:

Energy is conserved. According to the work-energy theorem, the work (W) done by friction force is equal to the change in mechanical energy (ME), which is the sum of potential energy (PE) and kinetic energy (KE). Since the surface is horizontal, the friction force is equal to the weight of the block (mg) times the coefficient of friction (μ). Initially, there is elastic potential energy (PE) stored in the spring, which is half the spring stiffness (k) times the square of the displacement (x). Finally, there is kinetic energy (KE) stored in the block, which is half the mass (m) times the square of the speed (v).

Work = Final mechanical energy − Initial mechanical energy

Fd = KE − PE

mgμd = ½ mv² − ½ kx²

½ mv² = ½ kx² + mgμd

v² = kx²/m + 2gμd

Plug in values and solve.

v² = (325 N/m) (0.50 m)²/(4.5 kg) + 2 (9.8 m/s²) (0.3) (5.0 m)

v = 6.9 m/s

Rounded to two significant figures, the final speed of the block is 6.9 m/s.

View image MathPhys

The block will come to rest due to the greater energy loss from friction than the initial potential energy stored in the spring, resulting in a final velocity of zero at the end of the surface.

First, let's determine the initial potential energy stored in the spring using Hooke's Law:

Potential Energy (PE) = (1/2)kx² = (1/2)(325 N/m)(0.50 m)² = 40.625 J.

Next, we'll find the work done by the friction force as the block slides. The friction force (f) is given by:

f = μmg = 0.3 × 4.5 kg × 9.8 m/s² = 13.23 N.

The work done by friction (Wfriction) over the distance (d) of 5.0 m is:

Wfriction = f × d = 13.23 N × 5.0 m = 66.15 J.

The total mechanical energy changes considering the work done by friction:

Initial PE - Wfriction = KEfinal.

So, 40.625 J - 66.15 J = KEfinal.

This simplifies to KEfinal = -25.525 J, indicating a loss of energy greater than the initial potential energy stored in the spring, implying the block would come to rest before reaching the end of the surface.

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