A block of mass 2 kg is propelled by a spring with spring constant k=5200N/m onto a smooth (frictionless) track. The spring is initially compressed a distance of 0.13 m from its equilibrium configuration. At the end of the track there is a rough inclined plane at an angle of 22 degrees with respect to the horizontal and with a coefficient of kinetic friction 0.15
a) How far up the incline will the block slide before coming to rest?
b) The coefficient of static friction is us= 0.7$. Will the block remain at rest on the incline? If not, how fast will it be going when it reaches the bottom again?
To solve these problems, we first need to calculate the initial energy stored in the compressed spring. Then we will calculate the work done against friction to find the distance the block slides up the incline and determine if it remains at rest or not. Finally, we will use the work-energy principle to calculate the speed of the block when it reaches the bottom of the incline.
a) To find how far up the incline the block will slide before coming to rest, we need to calculate the potential energy stored in the compressed spring and then find the work done against friction.
1. Calculate the potential energy stored in the compressed spring:
The formula for potential energy stored in a spring is given by:
Potential Energy = 0.5 * k * x^2
where k is the spring constant and x is the compression or extension of the spring from its equilibrium configuration.
Given:
Mass of the block (m) = 2 kg
Spring constant (k) = 5200 N/m
Compression of the spring (x) = 0.13 m
Substitute the given values into the formula:
Potential Energy = 0.5 * (5200 N/m) * (0.13 m)^2
Calculate:
Potential Energy = 0.5 * 5200 * 0.13^2
2. Calculate the work done against friction:
Since the track is smooth, there is no work done against friction on the track. Therefore, we only need to consider the work done against friction on the inclined plane.
The formula for work done against friction is given by:
Work = (Force of friction) * (Distance)
where Force of friction = coefficient of kinetic friction * normal force
and Distance is the distance moved along the inclined plane.
Given:
Angle of the incline (θ) = 22 degrees
Coefficient of kinetic friction (μ) = 0.15
Calculate:
Force of friction = μ * (Weight of the block)
Weight of the block = m * g, where g is the acceleration due to gravity (assume 9.8 m/s^2)
Substitute the values and calculate:
Force of friction = 0.15 * (2 kg * 9.8 m/s^2)
Distance = ?
3. Solve for the distance moved along the incline:
The work done against friction is equal to the change in potential energy stored in the spring. Thus, the work done against friction is equal to the initial potential energy.
Set the work done against friction equal to the potential energy calculated earlier and solve for the distance:
Work = Potential Energy
Force of friction * Distance = Potential Energy
Substitute the values and solve for Distance:
0.15 * (2 kg * 9.8 m/s^2) * Distance = 0.5 * 5200 * 0.13^2
Calculate:
Distance = (0.5 * 5200 * 0.13^2) / (0.15 * 2 * 9.8)
Therefore, the block will slide a distance given by the calculated value of Distance up the incline before coming to rest.
b) To determine if the block will remain at rest on the incline or not, we need to calculate the force of static friction exerted on the block. If the force of static friction is greater than or equal to the force trying to pull the block down the incline (its weight component), the block will remain at rest. Otherwise, it will start moving.
1. Calculate the force of static friction:
The force of static friction is given by:
Force of static friction = coefficient of static friction * normal force
Given:
Coefficient of static friction (μs) = 0.7
Calculate:
Force of static friction = μs * (Weight of the block)
2. Compare the force of static friction and the weight component:
The weight component down the incline is given by:
Weight component = Weight of the block * sin(θ)
Calculate:
Weight component = 2 kg * 9.8 m/s^2 * sin(22 degrees)
3. Compare the two forces:
If the force of static friction is greater than or equal to the weight component, the block will remain at rest. Otherwise, it will start moving.
Therefore, use the calculated values of the force of static friction and the weight component to determine if the block remains at rest or not.
If the block starts moving, we can use the work-energy principle to calculate its speed when it reaches the bottom of the incline. However, given the information in the problem, we are not able to determine this.