A spring with a spring-constant 2.2 N/cm is compressed 28 cm and released. The 5 kg mass skids down the frictional incline of height 31 cm and inclined at a 18◦ angle.
The acceleration of gravity is 9.8 m/s2 .
The path is frictionless except for a distance of 0.8 m along the incline which has a coefficient of friction of 0.3.
k = 2.2 N/cm
What is the final velocity vf of the mass? Answer in units of m/s
k = 2.2 N/cm =220 N/m. x =0.28 m, h=0.31 m, s =0.8 m, μ=0.3, α=18º.
PE(spring)=PE +W(fr) +KE,
k•x²/2=m•g•h+ μ•m•g•cos α•s +m•v²/2,
v=sqrt{kx²/m – 2g(h+ μ• s•cos α)}
Why did the spring go on a date? Because it wanted to have a spring fling!
To find the final velocity of the mass, we'll need to consider the forces acting on it. First, let's calculate the gravitational force:
Fg = m * g
Fg = 5 kg * 9.8 m/s^2
Fg = 49 N
Next, let's calculate the force applied by the spring when it's compressed:
Fspring = k * x
Fspring = 2.2 N/cm * 28 cm
Fspring = 61.6 N
Now, let's calculate the work done by the spring to compress the mass:
Wspring = (1/2) * k * x^2
Wspring = (1/2) * 2.2 N/cm * (28 cm)^2
Wspring = 862.4 N·cm = 8.624 N·m
The work done by the spring is equal to the decrease in potential energy:
Wspring = ∆PE
∆PE = m * g * h
8.624 N·m = 5 kg * 9.8 m/s^2 * 0.31 m
∆PE = 15.19 J
Now, let's consider the work done by friction over the inclined distance:
Wfriction = µ * m * g * d * cos(theta)
Wfriction = 0.3 * 5 kg * 9.8 m/s^2 * 0.8 m * cos(18°)
Wfriction = 5.478 J
The total work done on the mass is the sum of the work done by the spring and the work done by friction:
Wtotal = Wspring + Wfriction
Wtotal = 8.624 J + 5.478 J
Wtotal = 14.102 J
Finally, let's use the work-energy principle to find the final velocity:
Wtotal = ∆KE
14.102 J = (1/2) * m * vf^2
vf^2 = (2 * 14.102 J) / 5 kg
vf^2 = 8.9608 m^2/s^2
vf = sqrt(8.9608) m/s
vf ≈ 2.99 m/s
Therefore, the final velocity vf of the mass is approximately 2.99 m/s.
To find the final velocity (vf) of the mass, we need to consider the different stages of its motion.
1. Compression of the spring:
The potential energy stored in the compressed spring is given by:
Potential Energy (PE) = (1/2)kx^2,
where k is the spring constant (in N/cm) and x is the compression of the spring (in cm).
Substituting the given values:
PE = (1/2)(2.2 N/cm)(28 cm)^2 = 862.4 N•cm = 8.624 N•m.
2. Conversion of potential energy to kinetic energy:
When the spring is released, the potential energy is converted into kinetic energy.
Kinetic Energy (KE) = Potential Energy (PE) = 8.624 N•m.
3. Motion down the incline:
The total energy of the system will remain constant throughout the motion. This means that the kinetic energy gained will be equal to the work done against friction.
The gravitational potential energy lost as the mass slides down the incline is given by:
Potential Energy (PE) = mgh,
where m is the mass (in kg), g is the acceleration due to gravity (in m/s^2), and h is the height of the incline (in m).
Substituting the given values:
PE = (5 kg)(9.8 m/s^2)(0.31 m) = 15.145 N•m.
The work done against friction is given by:
Work = force × distance,
where force = u × Normal force and u is the coefficient of friction.
The normal force can be calculated as:
Normal force = m × g × cos(theta),
where theta is the angle of the incline.
Substituting the given values:
Normal force = (5 kg)(9.8 m/s^2)(cos(18°)) = 47.714 N.
The work done against friction is:
Work = (0.3)(47.714 N)(0.8 m) = 11.4292 N•m.
Since the total energy of the system remains constant, the kinetic energy gained is equal to the work done against friction:
Work = KE = 11.4292 N•m.
4. Finding the final velocity:
The kinetic energy gained is equal to (1/2)mv^2, where m is the mass and v is the final velocity.
Substituting the given values:
(1/2)(5 kg)(vf)^2 = 11.4292 N•m.
Simplifying the equation:
2.5(vf)^2 = 11.4292,
(vf)^2 = 4.57168,
vf ≈ √4.57168 ≈ 2.138 m/s.
Therefore, the final velocity (vf) of the mass is approximately 2.138 m/s.
To find the final velocity (vf) of the mass after skidding down the incline, we need to calculate the work done by the spring, the potential energy gained by the mass, and the work done against friction.
1. Calculate the work done by the spring:
The work done by a spring is given by the equation:
Work = (1/2) * k * x^2,
where k is the spring constant and x is the distance compressed.
In this case, k = 2.2 N/cm and x = 28 cm.
Converting k to N/m: 2.2 N/cm * (1 m/100 cm) = 0.022 N/m.
Converting x to meters: 28 cm * (1 m/100 cm) = 0.28 m.
Substituting the values, we get:
Work = (1/2) * 0.022 N/m * (0.28 m)^2.
Calculating that gives:
Work = 0.0024 J (joules).
2. Calculate the potential energy gained by the mass:
The potential energy gained by the mass is given by the equation:
Potential Energy = m * g * h,
where m is the mass, g is the acceleration due to gravity, and h is the height of the incline.
In this case, m = 5 kg, g = 9.8 m/s^2, and h = 31 cm.
Converting h to meters: 31 cm * (1 m/100 cm) = 0.31 m.
Substituting the values, we get:
Potential Energy = 5 kg * 9.8 m/s^2 * 0.31 m.
Calculating that gives:
Potential Energy = 15.19 J.
3. Calculate the work done against friction:
The work done against friction is given by the equation:
Work = frictional force * distance,
where the frictional force is the product of the coefficient of friction and the normal force.
In this case, the coefficient of friction is 0.3 and the distance is 0.8 m.
To calculate the normal force, we need to resolve the weight (mg) perpendicular to the incline.
Normal Force = mg * cos(angle),
where angle is the angle of the incline.
In this case, m = 5 kg and the angle is 18 degrees.
Converting the angle to radians: 18 degrees * (pi/180) = 0.314 radians.
Substituting the values, we get:
Normal Force = 5 kg * 9.8 m/s^2 * cos(0.314).
Calculating that gives:
Normal Force = 47.95 N.
The frictional force can be calculated as:
Frictional Force = coefficient of friction * Normal Force.
Substituting the values, we get:
Frictional Force = 0.3 * 47.95 N.
Calculating that gives:
Frictional Force = 14.38 N.
Now, we can calculate the work done against friction:
Work = 14.38 N * 0.8 m.
Calculating that gives:
Work = 11.504 J.
4. Calculate the final velocity:
The final velocity can be found using the principle of conservation of mechanical energy, which states that the total mechanical energy (the sum of kinetic and potential energy) remains constant in the absence of external forces (except friction).
So, the total mechanical energy is given by the sum of the initial potential energy (0 J, as the spring is compressed) and the work done by the spring (0.0024 J):
Total Mechanical Energy = 0 J + 0.0024 J = 0.0024 J.
The final kinetic energy is given by the work done by the spring plus the potential energy gained by the mass minus the work done against friction:
Final Kinetic Energy = Work + Potential Energy - Work against friction.
Substituting the values, we get:
Final Kinetic Energy = 0.0024 J + 15.19 J - 11.504 J.
Calculating that gives:
Final Kinetic Energy = 3.6864 J.
Finally, we can calculate the final velocity using the equation:
Final Kinetic Energy = (1/2) * m * vf^2,
where m is the mass and vf is the final velocity.
In this case, m = 5 kg.
Substituting the values, we get:
3.6864 J = (1/2) * 5 kg * vf^2.
Simplifying the equation and solving for vf, we get:
vf^2 = (2 * 3.6864 J) / 5 kg.
vf^2 = 1.10592 J / (kg * m^2/s^2).
vf^2 = 1.10592 m^2/s^2.
Taking the square root of vf^2, we get:
vf = √1.10592 m^2/s^2.
Calculating that gives:
vf = 1.050 m/s (rounded to three significant figures).
Therefore, the final velocity (vf) of the mass is approximately 1.050 m/s.