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 determine the maximum potential energy the block can have.
First, let's analyze the situation before the block reaches the inclined plane. The block is initially launched by the compressed spring, so the potential energy stored in the spring is converted into the block's kinetic energy.
The potential energy stored in a spring is given by the formula:
PE = (1/2) * k * x^2
where k is the force constant of the spring and x is the displacement of the spring from its equilibrium position.
In this case, the force constant of the spring is given as k = 600 N/m and the initial compression of the spring is given as x = 0.15 m.
Plugging in these values into the formula, we can calculate the potential energy stored in the spring as:
PE_spring = (1/2) * 600 N/m * (0.15 m)^2 = 13.5 J
This potential energy is converted entirely into the kinetic energy of the block as it moves along the horizontal surface before reaching the inclined plane.
The kinetic energy of an object is given by the formula:
KE = (1/2) * m * v^2
where m is the mass of the object and v is its velocity.
Since the block is moving horizontally along a frictionless surface, its velocity remains constant throughout its motion. Therefore, we can equate the potential energy stored in the spring to the kinetic energy of the block:
PE_spring = KE_block
Substituting the known values, we have:
13.5 J = (1/2) * 1.2 kg * v^2
Rearranging the equation, we can solve for the velocity of the block:
v^2 = (2 * 13.5 J) / 1.2 kg
v^2 = 22.5 m^2/s^2
v ≈ 4.74 m/s
Now, let's consider the block's motion on the inclined plane. As the block moves up the incline, it experiences two opposing forces: its weight (mg) acting downward and the kinetic friction force (fk) acting in the opposite direction. The work done by these forces will affect the block's change in height.
The work done by the weight of the block is given by:
W_weight = m * g * h
where m is the mass of the block, g is the acceleration due to gravity, and h is the change in height.
The work done by the friction force is given by:
W_friction = fk * d
where fk is the force of kinetic friction and d is the displacement of the block along the incline.
Since the block is moving up the inclined plane, the net work done on the block is equal to the change in its kinetic energy (from the horizontal motion to the vertical motion). Therefore, we can write:
W_net = ΔKE = KE_final - KE_initial
Since the block comes to rest at the maximum height, its final kinetic energy is zero. Therefore, the net work done on the block is equal to the negative of its initial kinetic energy.
W_net = -KE_initial
Substituting the formula for kinetic energy, we have:
W_net = -(1/2) * m * v^2
Since work (W) is defined as force (F) times displacement (d) multiplied by the cosine of the angle between them, we can write:
W_net = F * d * cos(θ)
The force acting along the incline is the component of the weight down the incline, which is given by:
F = m * g * sin(θ)
Therefore, we can rewrite the equation for net work as:
W_net = m * g * sin(θ) * d * cos(θ)
Since we are interested in the change in height (h), we can express the displacement (d) in terms of the angle (θ) and the change in height:
d = h / sin(θ)
Substituting this into the equation for net work, we have:
W_net = m * g * sin(θ) * (h / sin(θ)) * cos(θ)
W_net = m * g * h * cos(θ)
Equating this to the negative of the initial kinetic energy, we get:
-(1/2) * m * v^2 = m * g * h * cos(θ)
Simplifying and solving for h, we have:
h = -(1/2) * v^2 / (g * cos(θ))
Substituting the known values, we have:
h = -(1/2) * (4.74 m/s)^2 / (9.8 m/s^2 * cos(40˚))
h ≈ -8.62 m
The negative sign indicates that the change in height is downward. However, since we are interested in the maximum vertical height reached by the block, we take the absolute value of h.
Therefore, the maximum vertical height reached by the block is approximately 8.62 meters.