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What is the relationship between centripetal acceleration and velocity in uniform circular motion?
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Centripetal acceleration is proportional to the square of the velocity. If velocity doubles, centripetal acceleration quadruples.
How do you calculate the centripetal force acting on an object in circular motion?
Centripetal force is calculated using the formula: F_c = m * a_c = (m * v^2) / r, where m is mass, v is velocity, and r is the radius of the circle.
Explain the concept of normal force in relation to a hill or valley in circular motion.
At the bottom of a valley, the normal force is maximum and given by N = F_c + mg. At the top of a hill, it is minimum, calculated as N = mg - F_c.
How are velocity and period related in uniform circular motion when considering angular speed?
Velocity v = (2 * π * r) / T relates linearly with radius and inversely with period, describing the constant speed in the circular path.
What role does mass play in the formulas for both centripetal force and acceleration?
Mass appears as a multiplicative factor in centripetal force (F_c = m * a_c) but cancels out in the formula for centripetal acceleration, which is independent of mass.
Describe centripetal acceleration in terms of velocity and radius.
Centripetal acceleration, a_c, is defined as a_c = v^2 / r, indicating it depends on the square of velocity and inversely on radius.
Explain how centripetal acceleration can be derived using radius and period.
Centripetal acceleration can be expressed as a_c = (4 * π^2 * r) / T^2, showing that it depends on both radius and the square of the period.
What equation must be satisfied for an object to maintain contact on a circular path?
The condition N = mg - F_c >= 0 must be satisfied so that the object maintains contact, meaning normal force should be non-negative.
What is the relationship between tension components Tx and Ty in a horizontal circle?
The relationship is given by tan θ = T_y / T_x, where T_x = m * v^2 / r and T_y = mg.
How does the angle of incline affect tension in a horizontally spinning object at an angle?
Tension's angle components relate to centripetal force and gravitational force, affecting how an object maintains circular motion: T_x = centripetal, T_y = weight with θ determining ratio.
What condition would cause an object to lose contact with the ground at the top of a hill?
An object loses contact if the normal force at the top of the hill becomes less than zero (N < 0).
What happens to the tension at the bottom of a vertical circle?
At the bottom of a vertical circle, tension is greatest and equals the centripetal force plus the weight of the object: T = F_c + mg.
What determines the centripetal force in a horizontal circle where tension components are considered?
In a horizontal circle, the centripetal force is determined by T_x (horizontal tension component) as T_x = (m * v^2) / r, while T_y is equal to mg.
What formula is used to find frequency in the context of uniform circular motion?
The frequency, f, is calculated as the reciprocal of the period: f = 1/T.
How does the period (T) of circular motion relate to velocity?
Velocity in circular motion is given by the formula: v = (2 * π * r) / T, where r is the radius and T is the period of one complete revolution.
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