A solid cylinder with a radius of 5.08 cm starts from rest at the top of a 12m long ramp inclined 20.3 degrees above the horizontal. When it reaches the bottom of the ramp 3.25s later the cylinder has a final linear velocity of 7.38 m/s. What was the final velocity when it reached the bottom
To find the final velocity of the solid cylinder when it reached the bottom of the ramp, we can use the principles of kinematics.
First, we need to identify the known values:
- Initial velocity (u) = 0 m/s (as the cylinder starts from rest)
- Distance (s) = 12 m
- Time (t) = 3.25 s
- Angle of inclination (θ) = 20.3 degrees
- Final velocity (v) = ? (what we need to find)
Since the ramp is inclined above the horizontal, we can separate the gravitational force acting on the cylinder into two components: one parallel to the ramp (mg sinθ) and the other perpendicular to the ramp (mg cosθ). Here, m is the mass of the cylinder and g is the acceleration due to gravity.
As the cylinder rolls down the incline, the work done by the parallel component of gravity results in kinetic energy, while the perpendicular component does not contribute to the motion. Therefore, we can use the work-energy principle to solve this problem.
The work-energy principle states that the change in an object's kinetic energy is equal to the net work done on the object. Mathematically, it can be expressed as:
ΔKE = W_net
Since the initial velocity is 0, the change in kinetic energy is equal to the final kinetic energy. Therefore, the equation becomes:
1/2 * m * v^2 = W_net
The net work done is given by the force acting parallel to the ramp multiplied by the distance traveled (s). The force parallel to the ramp is mg sinθ. Hence, we have:
1/2 * m * v^2 = (mg sinθ) * s
Now, we need to find the mass (m). The formula for the mass of a solid cylinder is:
m = density * volume
The volume of a solid cylinder is given by:
volume = π * r^2 * h
Where r is the radius of the cylinder and h is the height (length) of the cylinder. In this case, since the cylinder is rolling, the height of the cylinder is the same as the length of the ramp (12 m).
Once we have the mass, we can substitute it back into the earlier equation to find the final velocity (v).
So, to find the final velocity of the cylinder when it reaches the bottom of the ramp, you'll need to follow these steps:
1. Calculate the mass of the cylinder using the formula m = density * volume, with volume = π * r^2 * h.
2. Substitute the mass value into the equation 1/2 * m * v^2 = (mg sinθ) * s.
3. Rearrange the equation to solve for v: v^2 = 2 * (mg sinθ * s) / m.
4. Take the square root on both sides to find the final velocity v.
By following these steps and plugging in the provided values, you should be able to calculate the final velocity of the cylinder when it reaches the bottom of the ramp.