A small block of mass 1.2 kg is launched by a compressed spring with force constant k=600 N/m. The initial compression of the spring is 0.15 m. The block slides along a horizontal frictionless surface and then up an inclined plane that makes an angle θ=40˚ with the horizontal. The coefficient of kinetic friction between the block and the inclined plane is μk=0.2. Use g=9.8m/s^2.
1. Find the maximum vertical height h reached by the block.
To find the maximum vertical height reached by the block, we need to analyze the energy of the system.
Initially, the block is at rest and all the energy is stored in the compressed spring. As the spring is released, it imparts kinetic energy to the block, which will then convert to potential energy as it moves up the inclined plane.
First, let's find the speed of the block when it reaches the top of the incline.
The potential energy stored in the compressed spring is given by:
PE_spring = (1/2)kx^2
where k is the force constant of the spring and x is the initial compression of the spring.
Substituting the given values, we have:
PE_spring = (1/2) * 600 N/m * (0.15 m)^2 = 13.5 J
This potential energy will be converted to kinetic energy when the block reaches the top of the incline. The kinetic energy is given by:
KE = (1/2)mv^2
where m is the mass of the block and v is its velocity. Since there is no friction along the horizontal surface, the kinetic energy remains constant as the block moves up the incline.
Next, let's consider the potential energy at the top of the incline, which is equal to the maximum potential energy reached by the block.
PE_top = mgh
where h is the maximum vertical height reached by the block.
We can equate the potential energy stored in the compressed spring to the potential energy at the top of the incline:
PE_spring = PE_top
13.5 J = mgh
Now, let's find the mass of the block. Given that m = 1.2 kg, we can rearrange the equation:
h = 13.5 J / (m * g)
h = 13.5 J / (1.2 kg * 9.8 m/s^2)
Calculating this, we find:
h ≈ 1.15 m
Therefore, the maximum vertical height reached by the block is approximately 1.15 meters.
To find the maximum vertical height reached by the block, we can use the principle of conservation of mechanical energy.
The initial mechanical energy of the block is stored in the compressed spring and can be calculated as the potential energy of the spring:
Initial potential energy (Uinitial) = (1/2) * k * x^2
where,
k = force constant of the spring (600 N/m)
x = initial compression of the spring (0.15 m)
Substituting the given values, we get:
Initial potential energy (Uinitial) = (1/2) * 600 * (0.15)^2
Next, we need to take into account the work done against friction as the block slides up the inclined plane. The work done against friction is given by:
Work done against friction (Wfriction) = μk * m * g * d
where,
μk = coefficient of kinetic friction (0.2)
m = mass of the block (1.2 kg)
g = acceleration due to gravity (9.8 m/s^2)
d = distance traveled up the incline
To find the distance traveled up the incline (d), we can use trigonometry. The inclined plane makes an angle of θ = 40˚ with the horizontal. The vertical distance traveled will be given by d * sin(θ).
d * sin(θ) = h
where,
h = maximum vertical height reached by the block
Substituting the values, we get:
d * sin(40˚) = h
Now, we can equate the initial potential energy of the block to the work done against friction and solve for h:
Uinitial = Wfriction
(1/2) * 600 * (0.15)^2 = 0.2 * 1.2 * 9.8 * d * sin(40˚)
Simplifying the equation:
(1/2) * 600 * 0.0225 = 2.352 * d * sin(40˚)
Now, solve for d:
d * sin(40˚) = (1/2) * 600 * 0.0225 / (2.352 * sin(40˚))
Finally, solve for h:
h = d * sin(40˚)
Evaluate the expression to find the value of h.