A small disc of mass 0.7 kg is attached to a string on a frictionless table top. The string passes through a small hole in the centre of the table and is attached to a mass of 0.68 kg suspended below the table and above the floor. The disc can thus move in a circular path with the hanging mass acting as a counterweight to keep the path radius (from the centre of the table) constant at 0.61 m. With what speed must the disc move, in m/s, around in its path such that the counterweight does not move?
Well, let's solve this problem step by step. First, let's find the tension in the string when the counterweight is not moving. Since the counterweight remains stationary, the tension in the string must be equal to its weight.
The weight of the counterweight can be calculated using the formula: weight = mass × acceleration due to gravity. In this case, the mass is 0.68 kg, and the acceleration due to gravity is approximately 9.8 m/s². So, the weight is:
weight = 0.68 kg × 9.8 m/s² ≈ 6.664 N
Now, to keep the disc moving in a circular path, the centripetal force (the force pulling the disc inward) must be provided by the tension in the string. The centripetal force can be calculated using the formula: centripetal force = mass × velocity² / radius.
In this case, the mass of the disc is 0.7 kg, the radius is 0.61 m, and we need to find the velocity. Rearranging the formula, we get:
velocity = sqrt(centripetal force × radius / mass)
Plugging in the given values, the formula becomes:
velocity = sqrt(6.664 N × 0.61 m / 0.7 kg) ≈ 3.279 m/s
So, the disc must move at a speed of approximately 3.279 m/s in its circular path for the counterweight to remain stationary.
To find the speed at which the disc must move in its circular path, we can use the concept of centripetal force.
The centripetal force required to keep the disc moving in a circular path is provided by the tension in the string. This tension is equal to the force of gravity acting on the hanging mass.
1. Calculate the force of gravity acting on the hanging mass:
Force of gravity = mass * acceleration due to gravity
Force of gravity = 0.68 kg * 9.8 m/s^2
Force of gravity = 6.664 N (approx.)
2. The tension in the string is equal to the force of gravity acting on the hanging mass.
Tension = Force of gravity = 6.664 N
3. The tension also provides the centripetal force to keep the disc in its circular path.
Centripetal force = Tension = 6.664 N
4. The centripetal force is given by the equation:
Centripetal force = (mass of the disc) * (velocity of the disc)^2 / (radius of the circular path)
Rearranging the equation we get:
(mass of the disc) * (velocity of the disc)^2 = (radius of the circular path) * (Centripetal force)
5. Substitute the given values into the equation:
0.7 kg * (velocity of the disc)^2 = 0.61 m * 6.664 N
6. Solve for the velocity of the disc:
(velocity of the disc)^2 = (0.61 m * 6.664 N) / 0.7 kg
(velocity of the disc)^2 = 5.8328 m^2/s^2
Taking the square root of both sides, we get:
velocity of the disc = √(5.8328 m^2/s^2)
velocity of the disc ≈ 2.414 m/s
Therefore, the disc must move at a speed of approximately 2.414 m/s in order to keep the counterweight from moving.
To find the speed at which the disc must move in its circular path, we can use the concept of centripetal force.
The counterweight hanging below the table provides the centripetal force required to keep the disc moving in a circular path. This force is given by the tension in the string. We can equate the tension in the string to the centripetal force.
The tension in the string can be calculated as the weight of the counterweight. The weight is equal to the mass of the counterweight multiplied by the acceleration due to gravity (9.8 m/s^2).
Weight of the counterweight = mass of the counterweight * acceleration due to gravity
= 0.68 kg * 9.8 m/s^2
Now, we can equate the tension in the string to the centripetal force required for the circular motion:
Tension = (mass of the disc * v^2) / r
Where:
Tension - tension in string
mass of the disc - mass of the small disc
v - speed of the disc
r - radius of the circular path
Solving this equation for the speed (v):
v^2 = (Tension * r) / mass of the disc
v = sqrt((Tension * r) / mass of the disc)
Now we can substitute the values into the equation to find the speed:
v = sqrt((0.68 kg * 9.8 m/s^2 * 0.61 m) / 0.7 kg)
Evaluating this expression:
v = sqrt(6.7474 m^2/s^2)
Therefore, the disc must move at a speed of approximately 2.60 m/s around its circular path in order to keep the counterweight from moving.