A small block slides along a track from one level to a higher level, by moving through an intermediate valley (see Figure). The track is frictionless until the block reaches the higher level. There a frictional force stops the block in a distance d. Assume that the block's initial speed is 10 m/s, the height difference h is 3.3 m, and μk is 0.87. Find the distance d that the block travels on the higher level before stopping.
To find the distance d that the block travels on the higher level before stopping, we need to use the principles of conservation of energy.
1. Firstly, let's calculate the potential energy (PE) of the block at the initial position (level 1).
- The formula for potential energy is: PE = m * g * h, where m is the mass of the block, g is the acceleration due to gravity, and h is the height difference.
- So, PE1 = m * g * h
2. Next, let's calculate the kinetic energy (KE) of the block at the initial position (level 1).
- The formula for kinetic energy is: KE = (1/2) * m * v^2, where m is the mass of the block and v is the initial velocity.
- So, KE1 = (1/2) * m * v^2
3. The total mechanical energy (E) of the block at the initial position (level 1) is the sum of its potential energy and kinetic energy.
- So, E1 = PE1 + KE1
4. At the higher level (level 2), the only form of energy left is potential energy.
- So, the potential energy at the higher level (PE2) is equal to the final mechanical energy (E2) minus the kinetic energy at the initial position (KE1).
- PE2 = E2 - KE1
5. At the higher level (level 2), the block comes to a stop due to friction. The work done by friction (Wf) is equal to the force of friction (f) multiplied by the displacement (d).
- The formula for work done is: Wf = f * d
- The force of friction (f) is equal to the coefficient of kinetic friction (μk) multiplied by the normal force (N).
- The normal force (N) is equal to the weight of the block (mg), where m is the mass of the block and g is the acceleration due to gravity.
- So, f = μk * (m * g), and Wf = μk * (m * g) * d
6. The work done by friction (Wf) is equal to the difference in potential energy at the higher level (PE2) and the potential energy at the initial position (PE1).
- So, Wf = PE2 - PE1
7. Equating the formulas for work done by friction (Wf) from step 5 and step 6, we can solve for d, the distance traveled on the higher level before stopping.
- μk * (m * g) * d = PE2 - PE1
- d = (PE2 - PE1) / (μk * (m * g))
Substituting known values into the equation will give you the distance d that the block travels on the higher level before stopping.