A solid cylinder rolls from rest down an incline 6m

high without slipping. calculate the linear velocity at the foot of the plane

mgh = 1/2 mv^2 + 1/2 I omega^2

You'll need to look up I for a solid cylinder (it will be a function of m and r^2) and for omega use v/r. The m's will all cancel, solve for v

To calculate the linear velocity at the foot of the plane for a solid cylinder rolling down without slipping, you can use the principle of conservation of energy.

1. Identify the given information:
- Height of the incline (h) = 6m

2. Calculate the potential energy at the top of the incline:
Potential energy (PE) = m * g * h

Where:
- m is the mass of the cylinder
- g is the acceleration due to gravity (approximately 9.8 m/s²)

3. Calculate the kinetic energy at the foot of the plane using the law of conservation of energy:
Kinetic energy (KE) = PE

Since the cylinder rolls without slipping, the total mechanical energy is conserved, meaning all of the potential energy is converted to kinetic energy.

4. Equate the expressions for potential and kinetic energy to find the linear velocity at the foot of the plane:
PE = KE

m * g * h = (1/2) * m * v²

Simplify the equation by cancelling out the mass (m) on both sides:

g * h = (1/2) * v²

Rearranging the equation:

v² = 2 * g * h

5. Take the square root of both sides to solve for the linear velocity (v):
v = √(2 * g * h)

Substitute the values of g (9.8 m/s²) and h (6m) into the equation:

v = √(2 * 9.8 m/s² * 6m)

v = √(117.6 m²/s²)

v ≈ 10.85 m/s

Therefore, the linear velocity at the foot of the incline is approximately 10.85 m/s.

To calculate the linear velocity of the solid cylinder at the foot of the inclined plane, we can use the principle of conservation of energy.

The potential energy gained by the cylinder as it rolls down the incline is equal to the kinetic energy it possesses at the bottom.

The potential energy gained by the cylinder can be calculated using the formula:

PE = m * g * h

where:
PE is the potential energy
m is the mass of the cylinder
g is the acceleration due to gravity (approximately 9.8 m/s^2)
h is the height of the incline (6m in this case)

Next, we can calculate the kinetic energy of the cylinder using the formula:

KE = (1/2) * m * v^2

where:
KE is the kinetic energy
m is the mass of the cylinder
v is the linear velocity (what we want to find)

Since the cylinder is rolling without slipping, the linear velocity can also be related to the angular velocity (ω) and the radius of the cylinder (r) using the equation:

v = ω * r

To obtain the solution, we need to relate the angular velocity to the linear velocity using the concept of rolling without slipping.

For a solid cylinder, the angular velocity (ω) is related to the linear velocity (v) as:

ω = v / r

Now, let's substitute the equations into each other to obtain the final solution:

PE = KE
m * g * h = (1/2) * m * v^2
g * h = (1/2) * v^2
2 * g * h = v^2
v = √(2 * g * h)

Plug in the given values into the equation for v:

v = √(2 * 9.8 * 6) = √(117.6) ≈ 10.84 m/s

Therefore, the linear velocity at the foot of the inclined plane is approximately 10.84 m/s.