A 6kilo gram package is placed on chat as shown at point A. the coefficient of friction between the package and and the chat is 0.15 . if the package is released from rest at A. determine the velocity of the package when it reaches the bottom of the chat and distance D from the bottom of the chat to point C where the package stops.
To determine the velocity of the package when it reaches the bottom of the chat, we can use the principles of conservation of energy. The package will undergo a loss in potential energy and a gain in kinetic energy as it moves from point A to the bottom of the chat.
1. Calculate the potential energy at point A:
Potential Energy (PE) = mass * gravity * height
PE = 6 kg * 9.8 m/s^2 * 0 m (since the height at point A is 0)
PE = 0 Joules
2. Calculate the kinetic energy at the bottom of the chat:
Kinetic Energy (KE) = (1/2) * mass * velocity^2
Since the package is released from rest, the initial velocity (v₀) is 0.
Therefore, KE = (1/2) * 6 kg * v^2
3. Apply the conservation of energy principle:
The loss in potential energy is equal to the gain in kinetic energy.
PE = KE
0 Joules = (1/2) * 6 kg * v^2
4. Solve for velocity (v):
0 = (1/2) * 6 kg * v^2
0 = 3 kg * v^2
v^2 = 0 m^2/s^2
v = 0 m/s
Therefore, the velocity of the package when it reaches the bottom of the chat is 0 m/s (since it already starts from rest).
Next, to determine the distance D from the bottom of the chat to point C where the package stops, we can calculate the net force acting on the package and use it to find the distance over which the package decelerates until it stops.
1. Calculate the net force:
Net Force (F_net) = Frictional Force (F_friction) - Weight (F_weight)
Frictional Force (F_friction) = coefficient of friction * Normal Force
Normal Force = mass * gravity
F_friction = 0.15 * (6 kg * 9.8 m/s^2)
F_friction = 8.82 N
Weight (F_weight) = mass * gravity
F_weight = 6 kg * 9.8 m/s^2
F_weight = 58.8 N
F_net = F_friction - F_weight
F_net = 8.82 N - 58.8 N
F_net = -49.98 N (negative sign indicates opposing direction)
2. Calculate the deceleration (a):
F_net = mass * acceleration
-49.98 N = 6 kg * a
a = -8.33 m/s^2 (again, negative sign indicates deceleration)
3. Calculate the distance (D):
Use the equation of motion: v^2 = v₀^2 + 2 * a * D
Since the package starts from rest at C, the final velocity (v) is 0.
Therefore, 0 = 0^2 + 2 * (-8.33 m/s^2) * D
Solving for D:
0 = -16.66 * D
D = 0 m
Therefore, the distance D from the bottom of the chat to point C where the package stops is 0 meters.
To determine the velocity of the package when it reaches the bottom of the chat and the distance D from the bottom of the chat to point C where the package stops, we can use the principles of mechanics, specifically Newton's laws of motion.
First, let's break down the problem into different stages:
1. Package released from rest at point A:
When the package is released from rest at point A, it will start moving downward due to gravity. At this stage, the package is not yet experiencing any frictional force.
2. Package starts experiencing friction:
As the package slides down the chat, it will eventually start experiencing friction with the chat. The frictional force can be determined using the formula: Frictional Force (Ff) = Coefficient of Friction (μ) * Normal Force (N), where the normal force is the force exerted by the chat on the package due to gravity.
To find the normal force, we need to calculate the weight of the package, which is the force acting vertically downward. The equation for weight (W) is: Weight (W) = mass (m) * acceleration due to gravity (g).
Given that the mass of the package is 6 kilograms and acceleration due to gravity is approximately 9.8 m/s^2, we can calculate the weight (W) as follows:
W = m * g
W = 6 kg * 9.8 m/s^2
W = 58.8 N
Now we can calculate the normal force (N) as the weight (W) acting vertically downward:
N = W
N = 58.8 N
Substituting this value into the frictional force (Ff) formula, we have:
Ff = μ * N
Ff = 0.15 * 58.8 N
Ff = 8.82 N
3. Determining the velocity and distance:
To determine the velocity of the package when it reaches the bottom of the chat and the distance D from the bottom of the chat to point C, we can analyze the forces acting on the package.
Considering the forces acting vertically, we have the weight (W) acting downward and the normal force (N) acting upward. The net force acting downward is the difference between these two forces, which is equal to W - Ff.
The net force acting on the package along the vertical direction can be expressed as: Net Force (Fnet) = mass (m) * acceleration (a), where the acceleration (a) can be represented as change in velocity (Δv) divided by time (t). Since the package is initially at rest, we can express the change in velocity (Δv) as the final velocity (v) minus the initial velocity (0). Hence, Δv = v - 0 = v. Therefore, the equation becomes:
Fnet = m * (v/t)
Using Newton's second law of motion, Fnet = m * a, we can equate the two expressions:
m * a = m * (v/t)
The mass (m) cancels out, giving us:
a = v/t
Alternatively, we can write the acceleration (a) as the change in velocity (Δv) divided by the change in time (Δt), which becomes:
a = Δv/Δt
Since the package is sliding downwards, the final velocity (v) can be expressed as the initial velocity (0) plus the change in velocity (v), i.e., v = 0 + v = v.
Using the above expressions, we have:
a = v/Δt
Now, considering the forces acting horizontally, we have the frictional force (Ff) acting to oppose the motion and no other significant forces. According to Newton's second law of motion, the net force acting on the package horizontally is equal to the frictional force (Ff):
Fnet = Ff
The net force acting on the package horizontally can be expressed as: Fnet = mass (m) * acceleration (a), where the acceleration (a) can be represented as the change in velocity (Δv) divided by time (t). Since the package is initially at rest, we can consider the change in velocity while moving from rest to the final velocity (v). Hence, Δv = v - 0 = v. Therefore, the equation becomes:
Ff = m * (v/t)
We can rearrange the equation to solve for time (t):
t = m * v / Ff
Substituting the given values: mass (m) = 6 kg, frictional force (Ff) = 8.82 N, and solving the equation, we should be able to find time (t).
Once we determine the time (t), we can use it to find the distance (D) using the equation for uniformly accelerated motion:
D = (1/2) * a * t^2
Substituting the values of acceleration (a) and time (t) we obtained earlier, we can solve for distance (D).
Finally, to find the velocity (v) when it reaches the bottom of the chat, we can use one of the equations of uniformly accelerated motion:
v = a * t
By substituting the values of acceleration (a) and time (t), we should be able to determine the velocity (v).