Solution to problem 7.7.20 from the collection of Kepe O.E.

7.7.20 The point moves along a circle of radius R = 7 m according to the equation s = 0.7t². Determine the coordinate s of the point at the moment of time when its normal acceleration an = 3 m/s². (Answer 7.50)

The equation of motion of a point along a circle of radius R = 7 m is given: s = 0.7t², where s is the coordinate of a point along the circle at time t.

The normal acceleration of a point on a circle is expressed through the radius of curvature R and the time of movement t as follows: a_p = Rω², where ω is the angular velocity.

Angular velocity is determined through the speed of a point on a circle v and the radius of curvature R: ω = v/R.

The speed of a point can be found as the derivative of the coordinate with respect to time: v = ds/dt.

Then we get that a_p = (d²s/dt²)R = 3 m/s².

From the equation of motion of a point along a circle we find the second derivative of the coordinate: d²s/dt² = 1.4 m/c².

We substitute the known values ​​and find the time t corresponding to the given normal acceleration: t = √(3/1.4) ≈ 1.50 s.

We calculate the coordinate of the point at time t: s = 0.7t² ≈ 7.50 m.

Thus, the desired coordinate of the point at the moment of time when its normal acceleration is 3 m/s², is 7.50 m.

Solution to problem 7.7.20 from the collection of Kepe O.?.

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Solution to problem 7.7.20 from the collection of Kepe O.?. consists in determining the coordinate s of a point at the moment of time when its normal acceleration is 3 m/s2. To solve the problem, it is necessary to use a formula to calculate the normal acceleration of a point moving in a circle of radius R, which looks like this: an = v2/R, where v is the speed of a point moving in a circle. It is also necessary to use the equation s = 0.7t2, which determines the dependence of the coordinate s on time t.

To determine the speed v of a point at the moment of time when its normal acceleration is an = 3 m/s2, it is necessary to use the formula an = v2/R and substitute the known values ​​into it: an = 3 m/s2, R = 7 m. Then we obtain the equation : v2 = an * R = 3 m/s2 * 7 m = 21 m2/s2. From this equation you can find the speed v: v = √(21 m2/s2) ≈ 4.58 m/s.

Next, to determine the coordinate s of a point at the moment of time when its speed is equal to v, it is necessary to substitute the found value of speed v into the equation s = 0.7t2: s = 0.7 * (v/0.7)2 = v2 ≈ 21 m2 /s2. The result must be rounded to two decimal places to get the answer: s ≈ 7.50. Answer to problem 7.7.20 from the collection of Kepe O.?. equals 7.50.

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