In the figure, a 4.3 kg block is accelerated from rest by a compressed spring of spring constant 660 N/m. The block leaves the spring at the spring's relaxed length and then travels over a horizontal floor with a coefficient of kinetic friction μk = 0.260. The frictional force stops the block in distance D = 7.8 m. What are (a) the increase in the thermal energy of the block–floor system, (b) the maximum kinetic energy of the block, and (c) the original compression distance of the spring?
To find the answers to the three questions, we'll need to break down the problem into smaller steps and use the relevant equations. Let's start with part (c), calculating the original compression distance of the spring.
To find the original compression distance of the spring, we can start by using Hooke's Law, which states that the force exerted by a spring is directly proportional to its displacement. The equation for Hooke's Law is:
F = -kx
Where F is the force exerted by the spring, k is the spring constant, and x is the displacement from the equilibrium position.
In this case, the force exerted by the spring is equal to the weight of the block (since it is at rest). Therefore, we can set up the equation:
mg = kx
Rearranging the equation to isolate x, we have:
x = mg / k
Now we can plug in the given values:
m = 4.3 kg (mass of the block)
g = 9.8 m/s^2 (acceleration due to gravity)
k = 660 N/m (spring constant)
x = (4.3 kg * 9.8 m/s^2) / 660 N/m
x ≈ 0.0648 m
So, the original compression distance of the spring is approximately 0.0648 meters.
Moving on to part (a), where we need to find the increase in the thermal energy of the block-floor system.
The increase in thermal energy can be calculated using the equation:
ΔE = μk * m * g * D
Where ΔE is the increase in thermal energy, μk is the coefficient of kinetic friction, m is the mass of the block, g is the acceleration due to gravity, and D is the stopping distance.
Plugging in the given values:
μk = 0.260
m = 4.3 kg
g = 9.8 m/s^2
D = 7.8 m
ΔE = 0.260 * 4.3 kg * 9.8 m/s^2 * 7.8 m
Calculating this expression, we find:
ΔE ≈ 87.162 J (approximately)
Therefore, the increase in the thermal energy of the block-floor system is approximately 87.162 joules.
Finally, let's move on to part (b) and find the maximum kinetic energy of the block.
Since the block comes to a stop due to friction, the work done by friction must be equal to the initial kinetic energy of the block. The work done by friction can be calculated using the equation:
Work = Force * distance
The force of friction can be found using the equation:
Force = μk * m * g
The distance over which the block is stopped is D = 7.8 m.
Therefore, the work done by friction is:
Work = μk * m * g * D
We know that the initial kinetic energy of the block is equal to the work done by friction:
Initial kinetic energy = Work
Plugging in the given values:
μk = 0.260
m = 4.3 kg
g = 9.8 m/s^2
D = 7.8 m
Initial kinetic energy = 0.260 * 4.3 kg * 9.8 m/s^2 * 7.8 m
Simplifying this expression, we find:
Initial kinetic energy ≈ 88.445 J (approximately)
Therefore, the maximum kinetic energy of the block is approximately 88.445 joules.
To summarize:
(a) The increase in the thermal energy of the block-floor system is approximately 87.162 joules.
(b) The maximum kinetic energy of the block is approximately 88.445 joules.
(c) The original compression distance of the spring is approximately 0.0648 meters.
m*g = 4.3kg * 9.8N/kg = 42.14 N. = Wt. of block.
b. a = u*g = 0.26 * (-9.8) = -2.548 m/s^2
V2^2 = V1^2 - 2*2.548*7.8 = 0
V1^2 = 39.75
V1 = 6.30 m/s = Initial velocity.
KE = m*V1^2/2 = 42.14*39.75/2 = 838 J.
c. = (1m/660N) * 42.14N = 0.0638 m.
(a) W=F*X
W= u*Fn*X
=>M*g*uk*X -> 4.3kg*0.26*7.8m
(b) same as (a)
(c) Us=0.5*k*x^2
=> (a)=0.5*640N/m*x^2
figure out X from that
[very late but someone might benefit from this]