A block of weight 3.8 N is launched up a 30 ∘ inclined plane of length 2.10 m by a spring with spring constant 2.05 kN/m and maximum compression 0.10 m . The coefficient of kinetic friction is 0.50.
To find the speed of the block when it reaches the top of the inclined plane, we can use the principle of conservation of mechanical energy. The mechanical energy of the system is conserved, meaning the initial potential energy (due to the spring compression) and the initial kinetic energy are equal to the final potential energy (at the top of the incline) and the final kinetic energy.
1. Calculate the potential energy at the start:
- The potential energy (PE) due to the spring, when it is compressed to its maximum, is given by: PE = (1/2)kx², where k is the spring constant and x is the maximum compression.
- Convert the spring constant from kN/m to N/m: 2.05 kN/m = 2050 N/m
- Substitute the values into the equation to find PE: PE = (1/2)(2050 N/m)(0.10 m)²
2. Find the work done against friction:
- The work done against friction is equal to the force of friction multiplied by the distance along the incline.
- Determine the force of friction using the coefficient of kinetic friction: f_friction = μ * m * g, where μ is the coefficient of kinetic friction, m is the mass of the block, and g is the acceleration due to gravity.
- Convert the weight of the block from N to kg: 3.8 N = 3.8 kg (since weight = mass * gravity)
- Determine the force of friction: f_friction = (0.50)(3.8 kg)(9.8 m/s²)
- Multiply the force of friction by the distance along the incline (2.10 m) to find the work done against friction.
3. Calculate the potential energy at the top of the incline:
- At the highest point, the block will have no kinetic energy. Therefore, the potential energy (PE_top) will be equal to the work done against friction plus the initial potential energy.
- PE_top = Work_done_against_friction + PE
4. Calculate the kinetic energy at the top of the incline:
- The kinetic energy of an object is given by: KE = (1/2) * m * v², where m is the mass of the block and v is the velocity of the block.
- Since all potential energy has been converted to kinetic energy, the kinetic energy at the top of the incline is equal to the potential energy at the top.
- Set the final potential energy equal to the kinetic energy equation and solve for v.
With these steps, you can calculate the velocity of the block when it reaches the top of the inclined plane.