A hollow cylinder (hoop) is rolling on a horizontal surface at speed v = 1.85 m/s when it reaches a 23.5° incline.How far up the incline will it go? How long will it be on the incline before it arrives back at the bottom?
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To find the distance the hollow cylinder will go up the incline, we can consider the conservation of mechanical energy.
1. Find the initial kinetic energy (KE_initial) of the hollow cylinder:
- The formula for kinetic energy is KE = 0.5 * m * v^2, where m is the mass of the cylinder and v is the speed.
- Since the cylinder is hollow, its mass is concentrated on the rim. The formula for the mass of a hoop is m = M * (2πr), where M is the mass per meter of the hoop and r is the radius of the hoop.
- Plug in the values to calculate the kinetic energy.
2. Find the potential energy at the maximum height (PE_max) of the hollow cylinder:
- The formula for potential energy is PE = m * g * h, where m is the mass, g is the acceleration due to gravity, and h is the height.
- At the maximum height, the cylinder loses all its kinetic energy, so the potential energy is equal to the initial kinetic energy.
- Rearrange the formula to solve for h (height).
3. Substitute the values and solve for h (height).
To find the time it takes for the hollow cylinder to arrive back at the bottom of the incline, we can use the conservation of mechanical energy and consider its motion along the incline.
4. Find the potential energy at the bottom of the incline (PE_bottom):
- Using the same formula as before, we can calculate the potential energy at the bottom of the incline.
5. Find the distance traveled along the incline (d):
- The formula for the distance traveled along an incline is d = s * cos(θ), where s is the distance traveled along the surface and θ is the angle of the incline.
- Calculate the distance traveled along the surface (s) using the formula s = v * t, where v is the initial speed and t is the time.
6. Find the time it takes to travel the distance (t):
- Rearrange the distance formula to solve for t.
7. Substitute the values and solve for t (time).
Let's now calculate the values step by step.
Step 1:
Given: v = 1.85 m/s, r = radius of the cylinder
Calculate: KE_initial = initial kinetic energy
Step 2:
Given: KE_max = KE_initial (at maximum height)
Calculate: PE_max = m * g * h (potential energy at maximum height)
Step 3:
Given: PE_max = KE_initial, g = acceleration due to gravity
Calculate: h = PE_max / (m * g) (height at maximum height)
Step 4:
Given: PE_bottom = PE_max (at bottom of incline)
Calculate: PE_bottom = m * g * h_bottom (potential energy at bottom of incline)
Step 5:
Given: θ = 23.5°, s = v * t, d = s * cos(θ)
Calculate: d = v * t * cos(θ) (distance traveled along the incline)
Step 6:
Given: d = v * t * cos(θ)
Calculate: t = d / (v * cos(θ)) (time traveled along the incline)
Step 7:
Substitute the known values into the formulas and calculated values to get the final answers.