A 5.40 Kg package slides 1.47 meters down a long ramp that is inclined at 11.6 degress below the horizontal. The coefficient of kinetic friction between the package and the ramp is = 0.313. Calculate the work done on the package by friction. Calculate the work done on the package by gravity. Calculate the work done on the package by the normal force. Calculate the total work done on the package. If the package has a speed of 2.15 m/s at the top of the ramp, what is its speed after sliding the distance 1.47 meters down the ramp?
component of weight normal to plane
Fn = m g cos 11.6 = 51.9 N
component of weight down the ramp
Fd = m g sin 11.6 = 10.7 N
Ff = Friction force = .313*Fn = 16.2 N
work done by friction = -1.47 Ff = -23.9 Joules (force opposite to motion so negative)
work done by gravity = m g * height change = m g * 1.47 sin 11.6
= 15.7 Joules
Normal force is perpendicular to motion so does no force
Total work
= 15.7 - 23.9 = - 8.2 Joules
Ke at end = Ke at start -8.2 Joule
(1/2) m v^2 = (1/2) m(2.15)^2 -8.2
.5 m v^2 = 4.28 Joules
v = 1.26 m/s
Well, well, well, we've got some work to do, don't we? Let's crunch some numbers, shall we?
First, let's calculate the work done by friction. The formula for work is W = F * d * cos(theta), where F is the force, d is the displacement, and theta is the angle between F and d. In this case, the force of friction is given by F = u * FN, where u is the coefficient of kinetic friction and FN is the normal force.
So, the work done by friction is W_f = (u * FN) * d * cos(theta).
Next, let's find the normal force. The normal force is given by FN = m * g * cos(theta), where m is the mass and g is the acceleration due to gravity.
Now, let's find the work done by gravity. The work done by gravity is simply the weight of the package multiplied by the displacement, and it can be calculated using the formula W_g = m * g * d * sin(theta).
Finally, let's calculate the total work done. The total work done is the sum of the work done by friction and the work done by gravity, so W_total = W_f + W_g.
Now, to find the final speed of the package after sliding down the ramp, we need to use the work-energy principle. The work done on an object is equal to the change in its kinetic energy. So, we can calculate the initial kinetic energy using the formula KE = (1/2) * m * v^2, where m is the mass and v is the initial speed. Since there is no external force doing work on the package, the work done by friction and gravity will be equal to the change in kinetic energy. So, we can write W_total = (1/2) * m * (v_final^2 - v_initial^2).
Now it's time to put all these formulas into action and get those answers. Let's get to work, shall we?
To solve this problem, we'll first calculate the work done on the package by friction, gravity, and the normal force. Then, we'll calculate the total work done on the package. Lastly, we'll calculate the final speed of the package after sliding down the ramp.
1. Work done by friction:
The work done by friction can be calculated using the formula:
Work done by friction = Force of friction * Distance
To find the force of friction, we need to calculate the normal force first:
Normal force (N) = Weight of the package (mg) * Cosine of the angle of the ramp (θ)
Weight of the package (mg) = mass of the package (m) * gravitational acceleration (g)
Gravitational acceleration (g) = 9.8 m/s^2
Weight of the package (mg) = 5.40 kg * 9.8 m/s^2 = 52.92 N
Cosine of the angle of the ramp (θ) = Cos(11.6°) = 0.977
Normal force (N) = 52.92 N * 0.977 = 51.68 N
Now we can calculate the force of friction:
Force of friction = Coefficient of kinetic friction * Normal force
Force of friction = 0.313 * 51.68 N = 16.16 N
Work done by friction = Force of friction * Distance
Work done by friction = 16.16 N * 1.47 m = 23.77 J
Therefore, the work done on the package by friction is 23.77 Joules.
2. Work done by gravity:
The work done by gravity can be calculated using the formula:
Work done by gravity = Weight of the package * Vertical distance
Vertical distance = Distance * Sine of the angle of the ramp (θ)
Sine of the angle of the ramp (θ) = Sin(11.6°) = 0.200
Vertical distance = 1.47 m * 0.200 = 0.294 m
Work done by gravity = Weight of the package * Vertical distance
Work done by gravity = 52.92 N * 0.294 m = 15.54 J
Therefore, the work done on the package by gravity is 15.54 Joules.
3. Work done by the normal force:
The work done by the normal force is zero because the direction of the normal force is perpendicular to the direction of motion. Therefore, the work done by the normal force is zero Joules.
4. Total work done on the package:
The total work done on the package is the sum of the work done by friction and gravity:
Total work done = Work done by friction + Work done by gravity
Total work done = 23.77 J + 15.54 J = 39.31 J
Therefore, the total work done on the package is 39.31 Joules.
5. Final speed of the package:
To calculate the final speed of the package, we'll use the law of conservation of energy.
The initial kinetic energy at the top of the ramp is equal to the final kinetic energy after sliding down the ramp.
Initial kinetic energy = 0.5 * mass * initial velocity^2
Final kinetic energy = 0.5 * mass * final velocity^2
Mass = 5.40 kg
Initial velocity = 2.15 m/s
Final velocity = ?
0.5 * 5.40 kg * (2.15 m/s)^2 = 0.5 * 5.40 kg * (final velocity)^2
24.47 J = 2.7 J * (final velocity)^2
(final velocity)^2 = 24.47 J / 2.7 J
(final velocity)^2 = 9.06
final velocity = √9.06 m/s
final velocity = 3.01 m/s
Therefore, the speed of the package after sliding the distance 1.47 meters down the ramp is approximately 3.01 m/s.
To calculate the work done on the package by friction, we can use the formula:
Work = Force * Distance
The force of friction can be calculated using the equation:
Force of friction = Coefficient of friction * Normal force
The normal force can be found using the equation:
Normal force = Mass * Gravity
where gravity is approximately 9.8 m/s^2.
First, let's calculate the force of friction:
Force of friction = 0.313 * (Mass * Gravity)
Next, we can calculate the work done by friction:
Work by friction = Force of friction * Distance
To calculate the work done on the package by gravity, we use the equation:
Work by gravity = Mass * Gravity * Distance * sin(angle)
where the angle is the inclination of the ramp in radians. To convert degrees to radians, we use the formula:
radians = degree * (pi/180)
Now, let's calculate the work done by gravity:
Work by gravity = Mass * Gravity * Distance * sin(angle)
To calculate the work done on the package by the normal force, we can use the equation:
Work by normal force = Normal force * Distance * cos(angle)
Finally, to calculate the total work done on the package, we add up the work done by friction, gravity, and the normal force:
Total work = Work by friction + Work by gravity + Work by normal force
To find the speed of the package after sliding down the ramp, we can use the principle of conservation of mechanical energy. The initial mechanical energy at the top of the ramp can be calculated using the formula:
Initial mechanical energy = (1/2) * Mass * (initial speed)^2
Since no work is done against gravity after sliding down the ramp, the final mechanical energy can be calculated as:
Final mechanical energy = (1/2) * Mass * (final speed)^2
Since mechanical energy is conserved, the initial mechanical energy is equal to the final mechanical energy, so we can set up the equation as follows:
(1/2) * Mass * (initial speed)^2 = (1/2) * Mass * (final speed)^2
Solving for the final speed, we can find:
Final speed = sqrt((initial speed)^2 + (2 * (Work by friction + Work by gravity + Work by normal force)) / Mass)