3- A 5.0 Kg block is released from point A in the figure below. It travels along the smooth surface A E except for the part B C where the surface is rough. Point A is 3 m above the lowest point C. The block continues in its motion up the surface until reaches point D (1.5 m above C) where it starts compressing the spring until it comes momentarily to rest at point E (∆x above point D). The speed of the block at point C is 6 m/s.
a) calculate the energy lost due to friction in the part B C;
b) if the spring constant kspring = 340 N/m, calculate the distance (∆x) by which the mass
compresses the spring just before it comes to rest momentarily.
To calculate the energy lost due to friction in the part B -> C, we need to find the work done by friction.
a) The work done by friction is given by the equation:
Work = Force * Distance * cos(theta)
Since the block is moving horizontally, the angle between the direction of motion and frictional force is 0 degrees.
Now, the work done by friction is equal to the energy lost due to friction:
Energy lost = Work
To find the force of friction, we can use the formula:
Force = mass * acceleration
The acceleration can be determined using Newton's second law of motion:
Force = mass * acceleration
Since the block is moving vertically, the only force acting on it in the vertical direction is the weight of the block:
Force = mass * acceleration = mass * gravity
Where gravity is approximately 9.8 m/s^2.
Now, let's calculate the force of friction:
Force of friction = mass * gravity = 5.0 kg * 9.8 m/s^2
Next, we need to determine the distance traveled in part B -> C. From the diagram, we know that point A is 3 m above point C.
Distance = 3 m
Finally, let's calculate the energy lost due to friction:
Energy lost = Force of friction * Distance * cos(theta)
Since the angle between the direction of motion and frictional force is 0 degrees, cos(theta) = 1.
Energy lost = Force of friction * Distance
Now, we can substitute the values to get the answer.
b) To calculate the distance (∆x) by which the mass compresses the spring just before it comes to rest momentarily, we can use the principle of conservation of mechanical energy.
At point D, the block has reached its maximum potential energy (PE) due to its highest position. This energy is converted into the potential energy stored in the spring (PE_spring) at point E.
Equating the potential energy equations, we can write:
m * g * ∆h = (1/2) * k * (∆x)^2
Where:
m = mass of the block = 5.0 kg
g = acceleration due to gravity = 9.8 m/s^2
∆h = height difference between point D and C = 1.5 m
k = spring constant = 340 N/m
∆x = distance by which the mass compresses the spring
Now, let's substitute the values and solve for ∆x:
5.0 kg * 9.8 m/s^2 * 1.5 m = (1/2) * 340 N/m * (∆x)^2
Simplify the equation and solve for ∆x.