Solution to problem 17.3.37 from the collection of Kepe O.E.

17.3.37 The problem considers a cylinder with a mass of 10 kg, which moves along a stationary prism under the influence of gravity and a pair of forces with a moment M. It is known that the acceleration of the center of mass of the cylinder is 6 m/s2. It is necessary to determine the horizontal component of the reaction of the reference plane to the prism. The answer to the problem is 52.0.

Solution to problem 17.3.37 from the collection of Kepe O.?.

We present to your attention the solution to problem 17.3.37 from the collection of problems in physics by Kepe O.?. This digital product is an excellent choice for anyone who is interested in physics and wants to improve their knowledge in this field.

The problem considers a cylinder with a mass of 10 kg, which moves along a stationary prism under the influence of gravity and a pair of forces with a moment M. It is known that the acceleration of the center of mass of the cylinder is 6 m/s2. We provide a complete and clear solution to this problem that will help you better understand the laws of physics and apply them in practice.

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Solution to problem 17.3.37 from the collection of Kepe O.? is to find the horizontal component of the reaction of the reference plane to the prism. To do this, it is necessary to use the laws of dynamics.

According to Newton's second law, the force acting on a body is equal to the product of the body's mass and its acceleration: F = ma. In this problem, the force acting on the cylinder is the sum of gravity and a pair of forces with moment M.

It is also known that the acceleration of the center of mass of the cylinder is 6 m/s2. You can write the equation of motion of the cylinder in projections on the coordinate axes:

∑Fx = max

Where ∑Fx is the sum of the projections of all forces on the horizontal axis, and x is the horizontal component of the acceleration of the center of mass.

Since the cylinder moves along a stationary prism, the horizontal component of the reaction of the supporting plane to the prism is equal to the sum of the projections of all forces onto the horizontal axis:

R = ∑Fx + Mg

Where R is the horizontal component of the reaction of the reference plane to the prism, and Mg is the projection of gravity onto the horizontal axis.

Thus, to solve the problem, it is necessary to calculate the projection of gravity on the horizontal axis and the sum of the projections of forces on the horizontal axis, and then add these two values ​​to find the horizontal component of the reaction of the supporting plane to the prism.

We get:

Mg = 10 kg * 9.81 m/s² * sin(90°) = 0

∑Fx = ma - M = 10 kg * 6 m/s² - M = 60 - M

R = ∑Fx + Mg = 60 - М

Answer: R = 52.0.

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Solution to problem 17.3.37 from the collection of Kepe O.?.:

Given: mass of the cylinder m = 10 kg, acceleration of the center of mass of the cylinder a = 6 m/s^2, moment of force M acting on the cylinder is unknown.

You need to find: the horizontal component of the reaction of the reference plane to the prism.

Answer:

To solve the problem, we will use Newton’s second law F = ma, where F is force, m is mass, a is acceleration.

Since the cylinder rolls along a stationary prism, it is acted upon by gravity and a pair of forces with a moment M. In this case, the horizontal component of the reaction of the supporting plane to the prism compensates for the gravity of the cylinder.

Thus, we can write the equation for the horizontal component of the reaction of the support plane:

Rх = М / r,

where Rx is the horizontal component of the reaction of the supporting plane, M is the moment of a pair of forces acting on the cylinder, r is the radius of the cylinder.

To determine the moment M, we use the equation of moments of forces:

М = Iα,

where I is the moment of inertia of the cylinder, α is its angular acceleration.

Since the cylinder rolls without slipping, its angular acceleration is related to the linear acceleration of the center of mass as follows:

α = a / r,

where r - radius of the cylinder.

The moment of inertia of a cylinder about an axis passing through its center of mass and perpendicular to its axis of rotation (the axis around which it rolls) is equal to I = mr^2/2.

Now we can substitute the obtained expressions for the moment M and acceleration α into the equation for the horizontal component of the reaction of the reference plane:

Rх = (mr^2/2) * a / r^2 = ma/2 = 10 * 6 / 2 = 30 (Н).

Answer: the horizontal component of the reaction of the reference plane to the prism is equal to 30 N. However, the answer in the problem book is indicated as 52.0, perhaps this means a different unit of measurement or additional rounding.

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