A 2.00kg solid sphere of radius 5.00cm rolls down a 20 degrees inclined plane starting from rest.
(a) What is the magnitude of the acceleration of the center of mass of the sphere?
(b) How far down the plane does it roll without slipping in 1.00s?
To find the answers to the given questions, we'll need to use some basic principles of physics, namely those related to rotational and translational motion.
(a) To determine the magnitude of the acceleration of the center of mass of the sphere, we'll need to consider the forces acting on the sphere.
Since the sphere is rolling down an inclined plane, it experiences two important forces: gravitational force (mg) and the normal force (N) exerted by the inclined plane. The normal force can be resolved into two components: one parallel to the incline (N⊥) and the other perpendicular to it (N⊥).
The gravitational force will have a component parallel to the incline (mg⊥) and a component perpendicular to it (mg⊥). The force of static friction (f_s) opposes the tendency of the sphere to slip down the incline.
Since the sphere is rolling without slipping, the static friction will provide a torque that accelerates the sphere's rotation and aids its linear acceleration. The static friction force (f_s) can be written as f_s = µ_s * N⊥, where µ_s is the coefficient of static friction.
Since the sphere is a solid sphere, the moment of inertia (I) is given by I = (2/5) * m * r^2, where m is the mass of the sphere and r is its radius.
The acceleration of the center of mass of the sphere can be found by using Newton's second law, which relates the net force on an object to its mass and acceleration: F_net = m * a_cm, where m is the mass of the sphere and a_cm is the acceleration of the center of mass.
So, to find the magnitude of the acceleration of the center of mass of the sphere, we need to consider the forces acting on the sphere and apply Newton's second law to rotational and translational motion simultaneously.
(b) To determine how far down the plane the sphere rolls without slipping in 1.00s, we can use the equations of motion for an object undergoing constant acceleration along an inclined plane.
We know that the initial velocity of the sphere is zero since it starts from rest. We also know the acceleration of the center of mass from part (a). The distance traveled on the inclined plane can be found using the following equation: s = v_0 * t + (1/2) * a * t^2, where s is the distance traveled, v_0 is the initial velocity, t is the time, and a is the acceleration.
By substituting the given values into the equation, we can find the distance traveled by the sphere without slipping in 1.00s.
Now, let's calculate the answers to the given questions using the given information.