A 3.0-kg sphere with a radius of 6.0 cm rolls from rest without slipping 4.0 m down an incline, reaching a speed of 7.0 m/s after rolling for 5.0 s. What is the torque that the friction interaction between the sphere and the incline exerts on the sphere? (Assume that the cylinder is moving toward you.)
To find the torque exerted by the friction interaction between the sphere and the incline, we can use the rotational dynamics equations. Let's break down the problem step by step:
Step 1: Find the moment of inertia (I) of the sphere.
The moment of inertia for a solid sphere rotating about its diameter is given by the formula:
I = (2/5) * m * r^2
where m is the mass of the sphere and r is its radius.
Using the given values, we can calculate the moment of inertia:
m = 3.0 kg
r = 6.0 cm = 0.06 m
I = (2/5) * 3.0 kg * (0.06 m)^2
Step 2: Calculate the angular acceleration (α) of the sphere.
The rotational kinetic energy (K_rot) of a rotating object is given by the formula:
K_rot = (1/2) * I * ω^2
where ω is the angular velocity.
Given that the sphere starts from rest and reaches a speed of 7.0 m/s after 5.0 s, we can calculate the angular velocity:
ω = Δθ / Δt = (7.0 m/s) / (0 s - 5.0 s)
where Δθ is the change in angular position and is equal to zero since the sphere starts from rest.
Substituting the values, we can calculate the angular acceleration:
α = ω / Δt
Step 3: Find the net torque (τ_net) on the sphere.
The net torque exerted on the sphere can be calculated using Newton's second law for rotation:
τ_net = I * α
Step 4: Calculate the torque exerted by friction (τ_friction).
The friction force (F_friction) can be determined by using the following equation:
F_friction = μ * m * g * cos(θ)
where μ is the coefficient of friction, m is the mass of the sphere, g is the acceleration due to gravity, and θ is the angle of the incline. However, since the problem does not provide the coefficient of friction or the angle of the incline, we cannot calculate the friction force directly.
Instead, we can use the relationship between torque and force:
τ_friction = r * F_friction
where r is the radius of the sphere in contact with the incline. In this case, it is equal to the radius of the sphere itself.
Therefore, to find the torque exerted by the friction interaction between the sphere and the incline, we need the coefficient of friction and the angle of the incline. Without those values, we cannot determine the exact torque.