A moving 3.20 kg block collides with a horizontal spring whose spring constant is 224 N/m.
The block compresses the spring a maximum distance of 5.50 cm from its rest position. The coefficient of kinetic friction between the block and the horizontal surface is 0.490. What is the work done by the spring in bringing the block to rest?
How much mechanical energy is being dissipated by the force of friction while the block is being brought to rest by the spring?
What is the speed of the block when it hits the spring?
To find the work done by the spring in bringing the block to rest, we can use the formula for the potential energy stored in a spring.
The potential energy stored in a spring that is compressed or stretched by a distance x from its equilibrium position is given by:
U = (1/2) k x^2
where U is the potential energy, k is the spring constant, and x is the distance from the equilibrium position.
In this case, the block compresses the spring by a maximum distance of 5.50 cm, which is equivalent to 0.055 m. The spring constant is given as 224 N/m.
Substituting these values into the formula, we can calculate the potential energy stored in the spring:
U = (1/2) * 224 N/m * (0.055 m)^2
Once we have calculated the potential energy stored in the spring, we can find the work done by the spring in bringing the block to rest. The work done by a force is equal to the change in potential energy associated with that force. Therefore, the work done by the spring is the negative of the change in potential energy:
Work done by the spring = -ΔU
Next, let's determine the amount of mechanical energy being dissipated by the force of friction while the block is being brought to rest.
Mechanical energy refers to the sum of kinetic energy and potential energy of the object. The change in mechanical energy can be calculated by subtracting the final mechanical energy from the initial mechanical energy:
Change in mechanical energy = Final mechanical energy - Initial mechanical energy
Since the block comes to rest, it means the final kinetic energy is zero. Therefore, the change in mechanical energy is equal to the initial kinetic energy plus the initial potential energy:
Change in mechanical energy = Initial kinetic energy + Initial potential energy
To find the initial kinetic energy, we need to calculate the initial speed of the block when it hits the spring. We can use the conservation of mechanical energy principle which states that the total mechanical energy of an object is conserved when only conservative forces are acting on it.
Since the only conservative force acting on the block is the spring force, the total mechanical energy can be calculated using the formula:
Total mechanical energy = Initial kinetic energy + Initial potential energy
Since the block starts from rest, its initial kinetic energy is zero. Therefore, the initial potential energy is equal to the initial mechanical energy.
Finally, to find the speed of the block when it hits the spring, we can use the conservation of mechanical energy principle. The total mechanical energy at the start, which is equal to the initial potential energy, is equal to the total mechanical energy when the block hits the spring, which is the sum of the final kinetic energy and the potential energy stored in the compressed spring. Using this information, we can calculate the final kinetic energy and then find the speed of the block.
I hope this explanation helps you solve the problem!
To find the work done by the spring in bringing the block to rest, we can use the equation for the potential energy stored in a spring:
Potential energy (U) = (1/2) * k * x^2
where k is the spring constant and x is the maximum distance the spring is compressed.
Given:
k = 224 N/m
x = 5.50 cm = 0.055 m
Plugging in the values, we can calculate the potential energy stored in the spring:
Potential energy (U) = (1/2) * (224 N/m) * (0.055 m)^2
= 0.361 Joules
Therefore, the work done by the spring in bringing the block to rest is 0.361 Joules.
To find the mechanical energy dissipated by the force of friction, we need to calculate the friction force and multiply it by the distance the block travels. The friction force can be found using the equation:
Friction force (f) = coefficient of friction (μ) * normal force (N)
where the normal force is equal to the weight of the block, given by:
Normal force (N) = mass (m) * acceleration due to gravity (g)
Given:
m = 3.20 kg
μ = 0.490
g = 9.8 m/s^2
Normal force (N) = (3.20 kg) * (9.8 m/s^2)
= 31.36 N
Friction force (f) = (0.490) * (31.36 N)
= 15.3544 N
The distance the block travels is equal to the maximum compression of the spring, which is 0.055 m as given. Therefore, the mechanical energy dissipated by the force of friction is:
Energy dissipated = force (f) * distance
= (15.3544 N) * (0.055 m)
= 0.844 Joules
The speed of the block when it hits the spring can be found using the principle of conservation of mechanical energy. The initial mechanical energy of the block is equal to its potential energy (due to its motion) plus the potential energy stored in the compressed spring. The final mechanical energy is equal to zero since the block comes to rest. Therefore, we can set up the equation:
Initial mechanical energy = final mechanical energy
(1/2) * m * (v^2) + U = 0
Substituting the values:
(1/2) * (3.20 kg) * (v^2) + 0.361 Joules = 0
Solving for v, we get:
(1/2) * (3.20 kg) * (v^2) = -0.361 Joules
(1.60 kg) * (v^2) = -0.361 Joules
v^2 = (-0.361 Joules) / (1.60 kg)
v^2 = -0.22625 m^2/s^2
Taking the square root of both sides, we find:
v = ± 0.475 m/s
Since the block is moving horizontally, the speed is expressed as a positive value. Therefore, the speed of the block when it hits the spring is 0.475 m/s.