A 320-N sphere 0.20 m in radius rolls without slipping 6.0 m down a ramp that is inclined at 25° with the horizontal. What is the angular speed of the sphere at the bottom of the slope if it starts from rest?
To find the angular speed of the sphere at the bottom of the slope, we can use the principle of conservation of energy. The initial potential energy of the sphere at the top of the slope will be converted into both kinetic energy and rotational kinetic energy at the bottom of the slope.
Here are the steps to find the answer:
Step 1: Determine the initial potential energy at the top of the slope.
To calculate the initial potential energy, use the formula:
Potential Energy = mass * gravity * height.
Given:
mass of the sphere (m) = unknown.
radius of the sphere (r) = 0.20 m.
acceleration due to gravity (g) = 9.8 m/s^2.
height of the slope (h) = unknown.
In this case, we need to find the height of the slope. We can use the sine function to relate the height and the angle of the slope:
sin(25°) = h / 6.0 m.
Solving for h:
h = 6.0 m * sin(25°).
Step 2: Calculate the mass of the sphere.
To find the mass of the sphere, we can use the formula:
Weight = mass * gravity.
Given:
weight of the sphere (W) = 320 N.
acceleration due to gravity (g) = 9.8 m/s^2.
Substituting the values, we get:
320 N = mass * 9.8 m/s^2.
Solving for mass:
mass = 320 N / 9.8 m/s^2.
Step 3: Calculate the initial potential energy.
Using the formula:
Potential Energy = mass * gravity * height,
Substituting the known values, we get:
Potential Energy = mass * 9.8 m/s^2 * h.
Step 4: Calculate the final kinetic energy and rotational kinetic energy.
At the bottom of the slope, the sphere will have both translational kinetic energy and rotational kinetic energy. The translational kinetic energy is given by:
Translational Kinetic Energy = (1/2) * mass * velocity^2.
Since the sphere is rolling without slipping, the velocity can be related to the angular velocity by:
velocity = radius * angular velocity.
Substituting the values, we get:
Translational Kinetic Energy = (1/2) * mass * (radius * angular velocity)^2.
The rotational kinetic energy is given by:
Rotational Kinetic Energy = (1/2) * moment of inertia * angular velocity^2.
For a solid sphere, the moment of inertia is given by:
Moment of Inertia = (2/5) * mass * radius^2.
Substituting the values, we get:
Rotational Kinetic Energy = (1/2) * ((2/5) * mass * radius^2) * angular velocity^2.
Step 5: Apply the conservation of energy principle.
According to the principle of conservation of energy:
Initial Potential Energy = Final Translational Kinetic Energy + Final Rotational Kinetic Energy.
Substitute the values and equations from the earlier steps into this equation:
mass * 9.8 m/s^2 * h = (1/2) * mass * (radius * angular velocity)^2 + (1/2) * ((2/5) * mass * radius^2) * angular velocity^2.
Step 6: Solve for the angular velocity.
Rearrange the equation to solve for the angular velocity:
((1/2) * mass * (radius * angular velocity)^2 + (1/2) * ((2/5) * mass * radius^2) * angular velocity^2) = mass * 9.8 m/s^2 * h.
Simplify and combine like terms:
((1/2) * (radius^2 + (4/5) * radius^2) * angular velocity^2) = 9.8 m/s^2 * h.
Simplify further:
((9/10) * radius^2 * angular velocity^2) = 9.8 m/s^2 * h.
Finally, solve for the angular velocity:
angular velocity^2 = (9.8 m/s^2 * h) / ((9/10) * radius^2).
Take the square root of both sides of the equation to find the angular velocity:
angular velocity = sqrt((9.8 m/s^2 * h) / ((9/10) * radius^2)).
Plug in the values for h and radius to calculate the angular velocity.