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.
Please I need the answer
To solve this problem, we will need to consider the conservation of mechanical energy. The mechanical energy of the system is conserved as long as no external forces other than gravity and friction are acting on the block.
Let's tackle the problem step by step:
a) To calculate the energy lost due to friction in the part B to C, we need to determine the work done by friction. The work done by friction is equal to the force of friction multiplied by the distance over which it acts.
First, we need to find the force of friction. We know that the block travels with a speed of 6 m/s at point C. The frictional force can be calculated using the equation:
Force of friction = mass × acceleration
Acceleration = (final velocity - initial velocity) / time
As the block starts compressing the spring at point D and comes to rest at point E, the final velocity at point C is 0 m/s. So, the acceleration can be calculated as follows:
Acceleration = (0 - 6) / (distance from C to D)
Acceleration = -6 / (1.5 m)
Next, we can find the force of friction:
Force of friction = mass × acceleration
Force of friction = 5.0 kg × (-6 / 1.5)
Force of friction = -20 N (negative sign indicates the opposite direction to motion)
Now, we need to find the distance over which the force of friction acts. The distance from B to C is given as 3 m.
Energy lost due to friction = Force of friction × distance from B to C
Energy lost due to friction = -20 N × 3 m
Energy lost due to friction = -60 J (negative sign indicates energy lost)
Therefore, the energy lost due to friction in the part B to C is 60 J.
b) To calculate the distance by which the mass compresses the spring just before it comes to rest momentarily (∆x), we need to consider the conservation of mechanical energy.
The mechanical energy of the system at point D is the sum of the gravitational potential energy and the energy stored in the spring.
Gravitational potential energy at point D = mass × g × height from C to D
Gravitational potential energy at point D = 5.0 kg × 9.8 m/s^2 × 1.5 m
The energy stored in the spring at point D is given by:
Energy stored in spring at point D = 0.5 × kspring × (∆x)^2
Since the block comes momentarily to rest at point E, all of the mechanical energy of the system is stored in the spring at that point (ignoring any energy lost to friction).
Equating the gravitational potential energy and the energy stored in the spring:
Mass × g × height from C to D = 0.5 × kspring × (∆x)^2
Plugging in the given values:
5.0 kg × 9.8 m/s^2 × 1.5 m = 0.5 × 340 N/m × (∆x)^2
Simplifying the equation:
73.5 kg·m^2/s^2 = 170 N/m × (∆x)^2
Solving for (∆x):
(∆x)^2 = (73.5 kg·m^2/s^2) / (170 N/m)
(∆x)^2 = 0.4324 m^2
Taking the square root of both sides:
∆x = sqrt(0.4324 m^2)
∆x ≈ 0.66 m
Therefore, the distance by which the mass compresses the spring just before it comes to rest momentarily is approximately 0.66 m.