2-A 10.0 Kg block is released from point A in the figure below. The block travels along a smooth frictionless track except for the rough 1 m long portion between points B and C. At the end of the track, the 10 kg block hits a spring of force constant (k) = 2000 N/m, and compresses the spring a distance of 20 cm from its equilibrium position before coming to rest momentarily.
a) What is the speed of the block at point B?
b) What is the speed of the block at point C?
c) What is the value of the coefficient of kinetic friction (k) between the block and the rough
portion between points B and C?
s=BC=1 m
a) PE=KE₁
mgh = mv₁²/2
v₁=sqrt(2gh)=…
b,c) KE₂=PE(spring)
mv₂²/2 =kx²/2
v₂=sqrt{ kx²/m} =…
KE₁=W(fr)+KE₂
mv₁²/2 = μmgs+ mv₂²/2
μ=(1/mgs)•{ mv₁²/2 - mv₂²/2}=…
0.328
To solve this problem, we need to consider the conservation of energy and the laws of motion.
a) To find the speed of the block at point B, we can use the principle of conservation of energy. At point A, the block has only potential energy, given by the equation:
PEA = mgh
where m is the mass of the block (10.0 kg), g is the acceleration due to gravity (9.8 m/s^2), and h is the height of point A above the ground.
Next, we calculate the potential energy at point B. Since the track between point B and C is smooth and frictionless, the block does not lose or gain any energy. Therefore, the potential energy at point B is equal to the potential energy at point A:
PEB = PEA = mgh
Since the height above the ground at point B is zero, the potential energy is also zero:
PEB = 0
Now, we can equate the two expressions for potential energy at points A and B:
mgh = 0
Simplifying the equation, we find:
gh = 0
Since the acceleration due to gravity is always positive, we can conclude that h must be zero. Therefore, the block is at ground level at point B.
At ground level, the block only has kinetic energy, which is given by:
KE = (1/2)mv^2
where v is the velocity of the block.
Setting the potential energy at point B equal to the kinetic energy, we have:
mgh = (1/2)mv^2
Simplifying and solving for v, we get:
v^2 = 2gh
v = sqrt(2gh)
Substituting the values given, we have:
v = sqrt(2 * 9.8 * 0)
Since the height is zero at point B, the velocity is also zero.
Therefore, the speed of the block at point B is 0 m/s.
b) To find the speed of the block at point C, we need to consider the conservation of mechanical energy. At point B, the block has potential energy (due to the spring) and kinetic energy. At point C, the block has only potential energy (due to the spring) because it comes to rest momentarily.
The potential energy at point B is given by:
PEB = (1/2)kx^2
where k is the force constant of the spring (2000 N/m), and x is the distance the spring is compressed (20 cm or 0.2 m).
Substituting the values, we have:
PEB = (1/2) * 2000 * (0.2)^2
PEB = 40 J
To find the velocity at point C, we set the potential energy at point B equal to the potential energy at point C:
PEB = PEC
(1/2)kx^2 = mgh
Simplifying and solving for v, we get:
v = sqrt(2gh)
Substituting the values, we have:
v = sqrt(2 * 9.8 * h)
Since the block comes to rest momentarily at point C, its velocity is zero. Therefore, the speed of the block at point C is also 0 m/s.
c) To find the coefficient of kinetic friction (μk) between the block and the rough portion between points B and C, we can use the equation:
mgμk = f
where m is the mass of the block (10.0 kg), g is the acceleration due to gravity (9.8 m/s^2), and f is the frictional force.
The frictional force can be calculated using the work-energy theorem:
f = ΔKE
where ΔKE is the change in kinetic energy between point B and point C.
From our previous calculations, we know that the kinetic energy at point B is zero, and the kinetic energy at point C is also zero. Therefore, the change in kinetic energy is:
ΔKE = 0 - 0 = 0 J
Substituting this value into the equation for the frictional force, we have:
mgμk = 0
Since the mass and acceleration due to gravity are always positive, we can conclude that μk must be zero.
Therefore, the value of the coefficient of kinetic friction (μk) between the block and the rough portion between points B and C is 0.