In the system shown below, the block π (mass 5.65 kg) is pushed so that it is initially moving to the left with a speed of π£0=1.38 m/s.
The hanging weight π has a mass of 2.86 kg and the coefficients of friction are ππ =0.411 and ππ=0.304. The string and the pulley have negligible mass. How fast will π be traveling when π has fallen through a height β=0.47 meters?
To solve this problem, we can apply the principles of Newton's laws of motion and the concept of work and energy.
Step 1: Analyze the forces acting on the system
First, let's identify the forces acting on the block π and the hanging weight π.
For the block π:
- The force due to gravity, ππ, acting downwards.
- The normal force, π, exerted by the surface, acting upwards.
- The friction force, π, opposing the motion of the block.
For the hanging weight π:
- The force due to gravity, ππ, acting downwards.
Step 2: Calculate the normal force and the friction force
The normal force is equal in magnitude and opposite in direction to the force due to gravity acting on the block. Thus, π = ππ.
The magnitude of the friction force, π, depends on the type of motion:
- If the block is not moving (static friction), the friction force can be calculated using the equation: π = ππ π, where ππ is the coefficient of static friction.
- If the block is already moving (kinetic friction), the friction force can be calculated using the equation: π = πππ, where ππ is the coefficient of kinetic friction.
Step 3: Calculate the acceleration of the system
Since the block π and the hanging weight π are connected by a string passing over a pulley and the system is assumed to be massless, the acceleration of both objects will be the same.
Using Newtonβs second law, we can write the equation: Ξ£πΉ = ππ, where Ξ£πΉ represents the net force acting on the system.
Using the forces we identified earlier, we can write the equation: ππ β π = ππ.
Solving for the acceleration, we get: π = (ππ β π) / π.
Step 4: Calculate the distance the hanging weight π falls
Given the height β through which π falls, we can determine the potential energy lost by the weight. The potential energy loss is given by: ππβ.
Step 5: Apply the work-energy principle
According to the work-energy principle, the work done on an object is equal to the change in its kinetic energy. Therefore, we can equate the work done against friction and the potential energy loss to the change in kinetic energy of the block.
From the work done against friction, we have: π Γ π = πππ Γ π = ππππ.
From the change in potential energy, we have: ππβ.
Equating the work done against friction and the potential energy loss, we get: ππππ = ππβ.
Step 6: Calculate the final velocity
Using the equation of motion, π£^2 = π£β^2 + 2ππ, we can determine the final velocity of the block π.
Rearranging the equation, we have: π£ = β(π£β^2 + 2ππ).
Substituting the values we calculated earlier for π and π, we can solve for π£.
Step 7: Perform the calculations
Now, let's substitute the given values into the equations and perform the calculations to find the final velocity π£.
Given:
π (mass of block π) = 5.65 kg
π (mass of hanging weight π) = 2.86 kg
ππ (coefficient of static friction) = 0.411
ππ (coefficient of kinetic friction) = 0.304
π£β (initial velocity) = 1.38 m/s
β (height fallen by hanging weight π) = 0.47 m
1. Calculate the normal force π: π = ππ = 5.65 kg Γ 9.8 m/s^2 = 55.37 N
2. Calculate the friction force π:
- If the block is not moving (static friction): π = ππ π = 0.411 Γ 55.37 N = 22.74 N
- If the block is already moving (kinetic friction): π = πππ = 0.304 Γ 55.37 N = 16.83 N
3. Calculate the acceleration π: π = (ππ β π) / π = (5.65 kg Γ 9.8 m/s^2 - 16.83 N) / 5.65 kg = 3.21 m/s^2
4. Calculate the distance π fallen by the hanging weight: π = β = 0.47 m
5. Calculate the final velocity π£:
π£ = β(π£β^2 + 2ππ) = β(1.38 m/s)^2 + 2 Γ 3.21 m/s^2 Γ 0.47 m = β(1.9044 + 3.0142) = β4.9186 = 2.22 m/s
Therefore, the block π will be traveling with a velocity of 2.22 m/s when the hanging weight π has fallen through a height of 0.47 meters.