A block of mass 0.49kg starts from rest at point A and slides down a frictionless hill of height h. At the bottom of the hill it slides across a horizontal piece of track where the coefficient of kinetic friction is 0.28. This section (from points B to C) is 3.60m in length. The block then enters a frictionless loop of radius r= 2.58m. Point D is the highest point in the loop. The loop has a total height of 2r.
What is the minimum kinetic energy for the block at point B in order to have enough speed at point D that the block will not leave the track?
What is the minimum height from which the block should start in order to have enough speed at point D that the block will not leave the track?
To calculate the minimum kinetic energy and the minimum height required for the block, we need to consider the conservation of energy and the conditions for the block to remain on the track at point D.
1. Minimum Kinetic Energy at Point B:
To find the minimum kinetic energy at point B, we need to consider the potential energy loss from point A to point B and the energy lost due to friction between points B and C.
a) Potential Energy Loss:
The potential energy at point A is given by PE(A) = mgh, where m is the mass of the block (0.49 kg) and g is the acceleration due to gravity (9.8 m/s²). Since the block starts from rest, all this potential energy is converted to kinetic energy at point B.
b) Energy Lost due to Friction between B and C:
The work done by friction can be calculated using the equation W = μk * N * d, where μk is the coefficient of kinetic friction (0.28), N is the normal force, and d is the distance traveled (3.60 m).
The normal force N can be calculated using the equation N = mg, where m is the mass of the block.
Therefore, the work done by friction W = μk * mg * d.
The total mechanical energy at point B is the sum of the kinetic energy and the potential energy loss, minus the work done by friction:
ME(B) = KE(B) + PE(B) - W
Since the block is initially at rest at point A, KE(B) = 0.
The potential energy loss from point A to point B is given by PE(A) - PE(B) = mgh - 0 = mgh.
Substituting these values into the equation, we have:
ME(B) = mgh - μk * mg * d
Simplifying further:
ME(B) = mg(h - μk * d)
2. Minimum Height at Point A:
To determine the minimum height required at point A for the block to have enough speed at point D, we need to consider the conservation of energy in the frictionless loop.
a) Potential Energy at Point D:
At the highest point of the loop (point D), the block will possess no kinetic energy since it momentarily comes to rest. Therefore, all the mechanical energy at point D is potential energy.
The potential energy at point D is given by PE(D) = mgh, where h is the vertical height of the loop (2r).
b) Total Mechanical Energy at Point A:
The total mechanical energy at point A is the sum of the potential energy, the kinetic energy, and the work done by friction:
ME(A) = KE(A) + PE(A) - W
As the block is at rest at point A, KE(A) = 0.
The potential energy at point A is given by PE(A) = mgh, where m is the mass of the block and g is the acceleration due to gravity.
Substituting these values into the equation, we have:
ME(A) = mgh - μk * mg * d
Simplifying further:
ME(A) = mg(h - μk * d)
Since the block needs to have enough speed at point D to remain on the track, the mechanical energy at point A should be equal to or greater than the potential energy at point D:
ME(A) ≥ PE(D)
Therefore, substituting the respective expressions:
mg(h - μk * d) ≥ mgh
Simplifying further:
h - μk * d ≥ h
h ≥ μk * d
Now we have the inequality that provides the minimum height required at point A to have enough speed at point D that the block does not leave the track.
To summarize:
1. Minimum Kinetic Energy at point B: ME(B) = mg(h - μk * d)
2. Minimum Height at point A: h ≥ μk * d
To determine the minimum kinetic energy at point B and the minimum height to start from, we need to consider the conservation of energy.
At point A, all the potential energy is converted into kinetic energy at the bottom of the hill (point B), and then all the kinetic energy is converted back into potential energy at the top of the loop (point D).
Let's break down the problem step-by-step:
Step 1: Calculate the potential energy at point B.
Potential energy at point B can be calculated using the formula:
Potential energy (PE) = mass (m) * gravity (g) * height (h)
Given:
mass (m) = 0.49 kg
gravity (g) = 9.8 m/s^2
height (h) = ?
Step 2: Calculate the kinetic energy at point B.
Kinetic energy at point B can be calculated using the formula:
Kinetic energy (KE) = 1/2 * mass (m) * velocity^2
Given:
mass (m) = 0.49 kg
velocity (v) = ?
coefficient of kinetic friction (μk) = 0.28
length of the track (l) = 3.60 m
Step 3: Calculate the velocity at point B using the work-energy principle.
The work done by friction (W) can be calculated using the formula:
Work (W) = force of friction (F) * distance (d)
The force of friction (F) can be calculated using the formula:
Force of friction (F) = coefficient of kinetic friction (μk) * normal force (Fn)
The normal force (Fn) can be calculated using the formula:
Normal force (Fn) = mass (m) * gravity (g)
Given:
mass (m) = 0.49 kg
gravity (g) = 9.8 m/s^2
coefficient of kinetic friction (μk) = 0.28
length of the track (l) = 3.60 m
Step 4: Calculate the minimum kinetic energy at point B using the velocity.
Kinetic energy at point B can be calculated using the formula:
Kinetic energy (KE) = 1/2 * mass (m) * velocity^2
Given:
mass (m) = 0.49 kg
velocity (v) = ?
coefficient of kinetic friction (μk) = 0.28
Step 5: Calculate the minimum height required to have enough speed at point D.
To determine the minimum height, we will equate the potential energy at point B to the potential energy at point D.
Potential energy at point D can be calculated using the formula:
Potential energy (PE) = mass (m) * gravity (g) * height (h)
Given:
mass (m) = 0.49 kg
gravity (g) = 9.8 m/s^2
height (h) = 2r
Step 6: Solve for height (h).
Now let's calculate step-by-step:
Step 1: Calculate the potential energy at point B.
Potential energy (PE) = mass (m) * gravity (g) * height (h)
PE = 0.49 kg * 9.8 m/s^2 * h
Step 2: Calculate the velocity at point B using the work-energy principle.
Work (W) = Force of friction (F) * distance (d)
F = μk * Fn
Fn = m * g
W = F * l
KE = W
KE = 1/2 * m * v^2
set KE = W,
1/2 * m * v^2 = μk * m * g * l,
We can cancel out mass, then
1/2 * v^2 = μk * g * l
v^2 = 2 * μk * g * l
v = sqrt(2 * μk * g * l)
Step 3: Calculate the minimum kinetic energy at point B using the velocity.
Kinetic energy (KE) = 1/2 * mass (m) * velocity^2
KE = 1/2 * 0.49 kg * (sqrt(2 * 0.28 * 9.8 m/s^2 * 3.60 m))^2
Step 4: Calculate the minimum kinetic energy at point B.
KE = 1/2 * 0.49 kg * (sqrt(2 * 0.28 * 9.8 m/s^2 * 3.60 m))^2 = 1.723 J
Therefore, the minimum kinetic energy at point B is 1.723 Joules.
Step 5: Calculate the minimum height required to have enough speed at point D.
Potential energy (PE) = mass (m) * gravity (g) * height (h)
PE = 0.49 kg * 9.8 m/s^2 * 2 * 2.58 m
Step 6: Solve for height (h).
h = (PE) / (m * g)
h = (0.49 kg * 9.8 m/s^2 * 2 * 2.58 m) / (0.49 kg * 9.8 m/s^2)
Therefore, the minimum height required to have enough speed at point D is 2.58 meters.