A 15.0 kg block is released from rest from a 3.00 m height at point A as shown in the figure below. The track is frictionless except for the portion between points B and C, which has a length of 4.20 m. The block travels down the track, hits at the end of it a
spring and compresses the spring 0.320 m from its equilibrium position before coming to rest momentarily. Then the spring expands, forcing the block back to the left, and the block - after losing the contact with the spring - continues sliding to the
left, slows down at the rough part of the track between C and B and stops exactly at
point B.
• A 3.00 m
B<————-4.20 m————->C
a) What is in magnitude the kinetic friction force, fr exerted on the block by the rough part of the track between points B and C?
Hint: use energy conservation either total or mechanical, whatever is applicable. "Before" is point A and "after" is point B as the block stops.
b) What is the spring constant, k?
Hint: use energy conservation again. Here you may choose "before" to be the point of the spring's maximum compression and "after" to be point B, again, as the block stops.
c) What is the speed of the block at point C?
Hint: use energy conservation again. Here you may choose "before" to be the point of the spring's maximum compression and "after" to be point C.
no answers?
If you use your calculator and if I use my sheets you'll get the right answer
a) To find the magnitude of the kinetic friction force exerted on the block by the rough part of the track between points B and C, we can use energy conservation.
At point A, the block has only gravitational potential energy (PE) given by:
PE = mgh
where m = 15.0 kg is the mass of the block, g = 9.8 m/s² is the acceleration due to gravity, and h = 3.00 m is the height.
At point B, the block has no more potential energy as it comes to rest, but it has kinetic energy (KE) due to its motion:
KE = 0.5mv²
where v is the velocity of the block at point B.
Since the track between B and C is rough and the block comes to rest, it means that all of its kinetic energy is converted into work done against friction.
Therefore, we can equate the initial potential energy to the work done by the frictional force:
mgh = fr * d
where fr is the magnitude of the kinetic friction force and d is the distance between B and C (4.20 m).
Solving for fr, we get:
fr = mgh/d
Substituting the given values, fr = (15.0 kg)(9.8 m/s²)(3.00 m)/(4.20 m), we can calculate the magnitude of the kinetic friction force.
b) To find the spring constant, k, we can again use energy conservation.
At the point of maximum compression of the spring, the block has no more potential energy but has potential energy stored in the compressed spring, given by:
PE = 0.5kx²
where k is the spring constant and x is the compression of the spring (0.320 m).
At point B, the block has no more potential energy or kinetic energy.
Equating the initial potential energy to the potential energy stored in the compressed spring, we have:
0.5kx² = mgh
where m, g, and h are the same as before.
Solving for k, we get:
k = 2mgh/x²
Substituting the given values, k = 2(15.0 kg)(9.8 m/s²)(3.00 m)/(0.320 m)², we can calculate the spring constant.
c) To find the speed of the block at point C, we can use energy conservation once again.
At the point of maximum compression of the spring, the block has potential energy stored in the compressed spring, as mentioned before.
At point C, the block has no more potential energy but has kinetic energy due to its motion:
KE = 0.5mv²
where v is the velocity of the block at point C.
Equating the initial potential energy stored in the compressed spring to the final kinetic energy at point C, we have:
0.5kx² = 0.5mv²
Solving for v, we get:
v² = kx²/m
Taking the square root of both sides, we get:
v = √(kx²/m)
Substituting the given values of k, x, and m, we can calculate the speed of the block at point C.