A solid sphere of mass m and radius r rolls without slipping down an inclined plane of angle θ. What is the acceleration of the center of mass of the sphere?
Practice Questions
1 question
Q1
A solid sphere of mass m and radius r rolls without slipping down an inclined plane of angle θ. What is the acceleration of the center of mass of the sphere?
g sin(θ)
g sin(θ)/2
g sin(θ)/3
g sin(θ)/4
The acceleration of the center of mass of a rolling object is given by a = g sin(θ) / (1 + k^2/r^2). For a solid sphere, k^2/r^2 = 2/5, thus a = g sin(θ) / (1 + 2/5) = g sin(θ) / (7/5) = (5/7)g sin(θ).
Questions & Step-by-step Solutions
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Q
Q: A solid sphere of mass m and radius r rolls without slipping down an inclined plane of angle θ. What is the acceleration of the center of mass of the sphere?
Solution: The acceleration of the center of mass of a rolling object is given by a = g sin(θ) / (1 + k^2/r^2). For a solid sphere, k^2/r^2 = 2/5, thus a = g sin(θ) / (1 + 2/5) = g sin(θ) / (7/5) = (5/7)g sin(θ).
Steps: 6
Step 1: Identify the forces acting on the sphere. The main force is gravity, which can be broken down into two components: one parallel to the incline (g sin(θ)) and one perpendicular to the incline (g cos(θ)).
Step 2: Understand that the sphere rolls without slipping. This means that as it rolls down, it also rotates. The acceleration of the center of mass (a) is related to the linear motion and the rotational motion.
Step 3: Use the formula for the acceleration of a rolling object: a = g sin(θ) / (1 + k^2/r^2), where k is the radius of gyration and r is the radius of the sphere.
Step 4: For a solid sphere, the value of k^2/r^2 is 2/5. Substitute this value into the formula: a = g sin(θ) / (1 + 2/5).
Step 5: Simplify the expression. The denominator becomes 1 + 2/5 = 7/5. So, a = g sin(θ) / (7/5).
Step 6: To simplify further, multiply by the reciprocal: a = (5/7) g sin(θ). This gives the final expression for the acceleration of the center of mass of the sphere.
