Question

An object moves in a circle of radius r at constant speed with a period t. If you want to change only the period in order to cut the object's acceleration in half, the new period should be?

1) t/2
2) 2t
3) t/4
4) 4t

Answer 1

To half the centripetal acceleration of an object moving in a circle, the period should be doubled. This is because the acceleration is inversely proportional to the square of the period.

Step-by-step explanation:

If an object moves in a circle of radius r at constant speed with a period t, its centripetal acceleration is given by the equation a = (4π²r)/t². To cut the object's acceleration in half, we need to adjust the period such that the new acceleration becomes a/2. Thus, we need the new period T to satisfy the equation (4π²r)/T² = (1/2)((4π²r)/t²). Solving for T gives us T = √t, which implies that the new period should be the square root of the original period squared. To simplify, the value of T that halves the acceleration would be 2t. Therefore, the correct answer to the question is option 2) 2t.

Answer 2

To halve the object's acceleration in circular motion, the period needs to be doubled, making the new period 2t.

2) 2t.

Step-by-step explanation:

To understand why the new period should be 2t to cut the object's acceleration in half, let's delve into the physics of circular motion. The acceleration a in circular motion is given by the formula
`\( a = (v^2)/(r) \)`, where v is the velocity and r is the radius of the circle. The period t is related to the velocity by the equation
`\( v = (2\pi r)/(t) \)`.

Now, if we want to cut the acceleration in half, we need to find the new period T that satisfies
`\( a_{\text{new}} = (v^2)/(r) = (1)/(2) \cdot (v^2)/(r) = (a)/(2) \)`. Substituting the expression for v in terms of t, we get
`\( a_{\text{new}} = (4\pi^2r)/(T^2) \)`.

Equating this to half of the original acceleration
`(\( a_{\text{original}} = (4\pi^2r)/(t^2) \))`, we find
`\( (4\pi^2r)/(T^2) = (1)/(2) \cdot (4\pi^2r)/(t^2) \)`. Solving for T, we get T = 2t. Therefore, the new period should be twice the original period to achieve the desired reduction in acceleration.