A solid sphere of mass 1 kg and radius 1 m rolls without slipping on a fixed inclined plane with an angle of inclination θ = 30° from the horizontal. Two forces of magnitude 1 N each, parallel to the incline, act on the sphere, both at distance r = 0.5 m from the center of the sphere, as shown in the figure. The acceleration of the sphere down the plane is ____ m/s². (Take g = 10m/s².)
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Answer:
To find the acceleration of the sphere down the plane, we can use the equations of rotational and translational motion.
Given:
- Mass of the sphere (m) = 1 kg
- Radius of the sphere (R) = 1 m
- Angle of inclination of the plane (θ) = 30°
- Magnitude of the forces (F) = 1 N each
- Distance from the center of the sphere to the points where the forces act (r) = 0.5 m
First, let's find the net torque acting on the sphere. Since the forces are parallel to the incline, they will create a torque that tends to rotate the sphere about its center. The net torque (τ) can be calculated as the difference between the torques created by the two forces:
τ = (Force 1) * (Lever arm 1) - (Force 2) * (Lever arm 2)
where:
- Lever arm 1 = distance from the center to the point where Force 1 acts = r = 0.5 m
- Lever arm 2 = distance from the center to the point where Force 2 acts = r = 0.5 m
τ = (1 N) * (0.5 m) - (1 N) * (0.5 m)
τ = 0 N.m
Since the net torque acting on the sphere is zero, there is no angular acceleration, and the sphere rolls without slipping.
Next, let's find the net force acting on the sphere along the incline. The component of the force parallel to the incline (F_parallel) can be calculated as:
F_parallel = F * sin(θ)
where θ is the angle of inclination (30°).
F_parallel = 1 N * sin(30°)
F_parallel = 0.5 N
Now, let's find the net force along the incline (F_net). The net force is the difference between the force parallel to the incline (F_parallel) and the component of the force due to gravity along the incline (mg_sin(θ)):
F_net = F_parallel - mg * sin(θ)
where m is the mass of the sphere (1 kg) and g is the acceleration due to gravity (approximately 9.8 m/s²).
F_net = 0.5 N - (1 kg) * (9.8 m/s²) * sin(30°)
F_net = 0.5 N - 4.9 N
F_net = -4.4 N
The net force along the incline is -4.4 N (negative because it acts opposite to the direction of motion).
Finally, we can use Newton's second law to find the acceleration (a) of the sphere down the incline:
F_net = m * a
-4.4 N = (1 kg) * a
a = -4.4 m/s²
The acceleration of the sphere down the inclined plane is -4.4 m/s². Since the acceleration is negative, it means that the sphere is moving down the plane with a deceleration of 4.4 m/s².
Explanation:
Answer:
The tendency of a rigid body to resist angular rotation is called moment of inertia. It is a quantity that depends on the shape of the body, mass distribution and orientation of the rotational axis. The ratio of angular momentum of a system to its angular velocity can also be called moment of inertia.