A ball of mass 0.1 kg is suspended vertically from the ceiling with a "massless" string and
brought to rest. The string is one meter long. When a force of 0.5N is applied 45 deg. below
the horizontal, what will be the magnitude of the resulting angular acceleration of the ball?
How would you go about solving this?
The torque is M=F•cosα•L.
The ball is material point => its moment of inertia is
I=m•L².
The Newton’s 2 law for rotation
M=I•ε,
F•cosα•L= m•L²•ε,
ε= F•cosα•L/ m•L²=F•cosα /m•L.
To solve this problem, we can use the concept of torque and rotational equilibrium.
Step 1: Identify the given information:
- Mass of the ball (m) = 0.1 kg
- Applied force (F) = 0.5 N
- Angle with the horizontal (θ) = 45 degrees
- Length of the string (r) = 1 meter
Step 2: Calculate the torque applied by the force:
The torque (τ) is given by the formula τ = r * F * sin(θ). Plugging in the values, we have:
τ = 1 meter * 0.5 N * sin(45 degrees)
Step 3: Calculate the moment of inertia:
The moment of inertia (I) for a solid sphere is given by the formula I = (2/5) * m * r^2, where m is the mass and r is the radius. Since we know the mass, we can calculate the radius using the formula r = (4/3) * π * (r^3) / m.
Plugging in the values, we have:
r = (4/3) * π * (0.1 kg / (4/3) π )^(1/3)
= (0.1 kg / (4/3) π )^(1/3)
≈ 0.2388 meters
Now, using the radius value we calculated, we can find the moment of inertia:
I = (2/5) * 0.1 kg * (0.2388 meters)^2
Step 4: Calculate the angular acceleration:
The angular acceleration (α) is given by the equation τ = I * α. Rearranging the equation, we have:
α = τ / I
Substituting the values we calculated for torque (τ) and moment of inertia (I), we can find the angular acceleration (α).
Step 5: Calculate the magnitude of the angular acceleration:
Plug the values calculated in the previous steps into the equation and evaluate to find the magnitude of the angular acceleration.
By following these steps, you can find the magnitude of the resulting angular acceleration of the ball.
To solve this problem, we need to apply the principles of torque and rotational motion. The torque acting on an object is equal to the product of the force applied and the perpendicular distance from the force to the axis of rotation. The angular acceleration is related to the torque through the moment of inertia of the object.
Here's how you can approach solving this problem step by step:
1. First, let's calculate the torque acting on the ball. The force applied is 0.5N, and it acts at an angle of 45 degrees below the horizontal. Since the string is one meter long, the perpendicular distance from the force to the axis of rotation is the horizontal component of the length of the string. We can calculate it using trigonometry:
- Distance = length of the string * sin(angle)
- Distance = 1m * sin(45 degrees)
2. Next, we need to find the moment of inertia of the ball. For a simple pendulum, the moment of inertia is equal to the mass of the object times the square of the distance from the axis of rotation (in this case, the ball's center of mass) to the axis of rotation (the pivot point in the ceiling):
- Moment of inertia = mass * distance^2
- Moment of inertia = 0.1kg * (distance)^2
3. Now, we can calculate the torque by multiplying the force and the perpendicular distance:
- Torque = force * distance
4. To find the angular acceleration, we use the formula:
- Torque = moment of inertia * angular acceleration
5. Rearranging the formula, we get:
- Angular acceleration = Torque / Moment of inertia
6. Substitute the calculated values of torque and moment of inertia into the formula to find the angular acceleration.
By following these steps, you should be able to find the magnitude of the resulting angular acceleration of the ball.