a 3.5 kg block is accelerated from rest by a compressed spring of spring constant 640 N/m. The block leaves the spring at the springs relaxation length.The block then travels over a horizontal floor with coefficient of kinetic friction 0.25. The friction force stops the block in a distance D = 7.8 m.
What is the increase in the thermal energy of the block+floor system?
(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]
You can calculate the spring compression X from
66.9 J = (1/2)kX^2, but you don't need it. 66.9 J is the thermal energy increease.
They gave you more information than you need to answer the question
All of the stored potential energy of the spring, (1/2)kX^2, winds up as heat. X is the spring compression, which they don't tell you.
You can compute the heat generated another way:
Friction work = M*g*Uk*X = 66.9 J
What is the original compression distance of the spring?
Thought it would be zero but that isn't right.
To find the increase in thermal energy of the block+floor system, we need to calculate the work done on the block by both the spring and friction forces. Since the block starts from rest, the work done by the spring will be equal to the increase in kinetic energy of the block, while the work done by friction will be equal to the decrease in kinetic energy of the block.
First, let's calculate the work done by the spring force:
The spring force can be calculated using Hooke's Law: F = -kx, where F is the force, k is the spring constant, and x is the displacement from the equilibrium position. In this case, the block leaves the spring at its relaxation length, which means the displacement is zero. Therefore, the spring force is also zero, and no work is done by the spring.
Next, let's calculate the work done by the friction force:
The friction force can be calculated using the formula: f = μN, where f is the friction force, μ is the coefficient of kinetic friction, and N is the normal force. The normal force can be calculated as the weight of the block, which is mass times gravity: N = mg, where m is the mass and g is the acceleration due to gravity.
Given that the mass of the block is 3.5 kg, the coefficient of kinetic friction is 0.25, and the acceleration due to gravity is 9.8 m/s^2, we can calculate the normal force:
N = (3.5 kg)(9.8 m/s^2) = 34.3 N
Now, we can calculate the friction force:
f = (0.25)(34.3 N) = 8.58 N
The work done by the friction force can be calculated using the formula: work = force x distance. In this case, the work done by friction will be the force of friction multiplied by the distance traveled, which is 7.8 m:
work = (8.58 N)(7.8 m) = 66.924 J
Finally, the increase in thermal energy of the block+floor system is equivalent to the work done by the friction force:
increase in thermal energy = 66.924 J
Therefore, the increase in thermal energy of the block+floor system is 66.924 Joules.