Chapter 7

A small block with mass 0.0400 kg slides in a vertical circle of radius \(R =\) 0.500 m on the inside of a circular track. During one of the revolutions of the block, when the block is at the bottom of its path, point \(A\), the normal force exerted on the block by the track has magnitude 3.95 N. In this same revolution, when the block reaches the top of its path, point \(B\), the normal force exerted on the block has magnitude 0.680 N. How much work is done on the block by friction during the motion of the block from point \(A\) to point \(B\)?

A small block with mass 0.0500 kg slides in a vertical circle of radius \(R =\) 0.800 m on the inside of a circular track. There is no friction between the track and the block. At the bottom of the block's path, the normal force the track exerts on the block has magnitude 3.40 N. What is the magnitude of the normal force that the track exerts on the block when it is at the top of its path?

A small rock with mass 0.12 kg is fastened to a massless string with length 0.80 m to form a pendulum. The pendulum is swinging so as to make a maximum angle of 45\(^\circ\) with the vertical. Air resistance is negligible. (a) What is the speed of the rock when the string passes through the vertical position? What is the tension in the string (b) when it makes an angle of 45\(^\circ\) with the vertical, (c) as it passes through the vertical?

A wooden block with mass 1.50 kg is placed against a compressed spring at the bottom of an incline of slope 30.0\(^\circ\) (point \(A\)). When the spring is released, it projects the block up the incline. At point \(B\), a distance of 6.00 m up the incline from A, the block is moving up the incline at 7.00 m/s and is no longer in contact with the spring. The coefficient of kinetic friction between the block and the incline is \(\mu_k =\) 0.50. The mass of the spring is negligible. Calculate the amount of potential energy that was initially stored in the spring.

A small object with mass \(m =\) 0.0900 kg moves along the \(+x\)-axis. The only force on the object is a conservative force that has the potential-energy function \(U(x) = -ax^2 + bx^3\), where \(\alpha =\) 2.00 J/m\(^2\) and \(\beta =\) 0.300 J/m\(^3\). The object is released from rest at small \(x\). When the object is at \(x =\) 4.00 m, what are its (a) speed and (b) acceleration (magnitude and direction)? (c) What is the maximum value of \(x\) reached by the object during its motion?

A cutting tool under microprocessor control has several forces acting on it. One force is \(\overrightarrow{F}\) \(= - \alpha xy^2 \hat\jmath\), a force in the negative \(y\)-direction whose magnitude depends on the position of the tool. For \(a =\) 2.50 N/m\(^3\), consider the displacement of the tool from the origin to the point (\(x =\) 3.00 m, \(y =\) 3.00 m). (a) Calculate the work done on the tool by \(\overrightarrow{F}\) if this displacement is along the straight line \(y = x\) that connects these two points. (b) Calculate the work done on the tool by \(\overrightarrow{F}\) if the tool is first moved out along the \(x\)-axis to the point (\(x =\) 3.00 m, \(y =\) 0) and then moved parallel to the y-axis to the point (\(x =\) 3.00 m, \(y =\) 3.00 m). (c) Compare the work done by \(\overrightarrow{F}\) along these two paths. Is \(\overrightarrow{F}\) conservative or nonconservative? Explain.

A proton with mass \(m\) moves in one dimension. The potential-energy function is \(U(x) = (\alpha/x^2) - (\beta/x)\), where \(\alpha\) and \(\beta\) are positive constants. The proton is released from rest at \(x_0\) = \(\alpha\)/\(\beta\). (a) Show that \(U(x)\) can be written as $$U(x) = \frac{\alpha}{x^2_0}\ \Big[ \Big( \frac{x_0}{x}\ \Big)^2 - \frac{x_0}{x}\ \Big] $$ Graph \(U(x)\). Calculate \(U(x_0)\) and thereby locate the point \(x_0\) on the graph. (b) Calculate \(v(x)\), the speed of the proton as a function of position. Graph \(v(x)\) and give a qualitative description of the motion. (c) For what value of \(x\) is the speed of the proton a maximum? What is the value of that maximum speed? (d) What is the force on the proton at the point in part (c)? (e) Let the proton be released instead at \(x_1 = 3\alpha / \beta\). Locate the point \(x_1\) on the graph of \(U(x)\). Calculate \(v(x)\) and give a qualitative description of the motion. (f) For each release point (\(x = x_0\) and \(x = x_1\)), what are the maximum and minimum values of \(x\) reached during the motion?

During the calibration process, the cantilever is observed to deflect by 0.10 nm when a force of 3.0 pN is applied to it. What deflection of the cantilever would correspond to a force of 6.0 pN? (a) 0.07 nm; (b) 0.14 nm; (c) 0.20 nm; (d) 0.40 nm.