A sphere of radius 20 cm and mass 1.80 kg starts from rest and rolls without slipping down a 30.0 degree incline...
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To calculate the sphere's acceleration as it rolls down the incline, we can use a combination of Newton's second law and the rotational equations of motion.
Step 1: Determine the gravitational force acting on the sphere.
The gravitational force can be calculated using the formula:
F_gravity = m * g
where m is the mass of the sphere and g is the acceleration due to gravity (approximately 9.8 m/s^2).
In this case, m = 1.80 kg, so the gravitational force is:
F_gravity = 1.80 kg * 9.8 m/s^2 = 17.64 N
Step 2: Resolve the force of gravity into its parallel and perpendicular components.
The force of gravity can be decomposed into a component parallel to the incline (F_parallel) and a component perpendicular to the incline (F_perpendicular).
F_parallel = F_gravity * sin(theta)
F_perpendicular = F_gravity * cos(theta)
where theta is the angle of the incline (30 degrees in this case).
Step 3: Calculate the net force acting on the sphere.
Since the sphere is rolling without slipping, the force accelerating it down the incline is the force of static friction (F_friction) between the sphere and the incline.
The net force can be calculated as:
F_net = F_parallel - F_friction
Step 4: Determine the moment of inertia of the rolling sphere.
For a solid sphere rolling without slipping, the moment of inertia (I) is given by:
I = (2/5) * m * r^2
where r is the radius of the sphere (20 cm or 0.2 m in this case).
Step 5: Use the rotational equations of motion.
Using the relation between torque (τ), angular acceleration (α), and moment of inertia (I) τ = I * α, we can relate the net force and the acceleration of the sphere.
τ = F_net * R
α = a/R
where R is the radius of the sphere and a is the linear acceleration of the sphere.
Step 6: Relate the linear acceleration and angular acceleration.
Since the sphere is rolling without slipping, the linear acceleration a can be related to the angular acceleration α by:
a = α * R
Step 7: Combine the equations and solve for acceleration.
Substituting the forces and moments, we have:
(F_parallel - F_friction) * R = (2/5) * m * r^2 * α * R
Simplifying, we obtain:
F_parallel - F_friction = (2/5) * m * r^2 * α
Step 8: Solve for acceleration (a).
Since F_parallel = F_gravity * sin(theta), we rewrite the equation as:
F_gravity * sin(theta) - F_friction = (2/5) * m * r^2 * α
Substituting the expression for F_friction, we have:
F_gravity * sin(theta) - μ_s * F_perpendicular = (2/5) * m * r^2 * α
where μ_s is the coefficient of static friction.
However, we are missing the value of the coefficient of static friction (μ_s), which is necessary to determine the acceleration of the sphere. Could you provide that information?